\section{Introduction}
\indent Given a simple compact connected Lie group $K$ and a maximal torus $T$ of $K$, the Weyl group $W:=N_K(T)/T$ acts on $T$ by conjugation by a representative element of $N_K(T)$. This is well-defined since $T$ is abelian. As $T$ is clearly a CW-complex, it is known that there exists a $W$-equivariant cellular structure on $T$. The first aim of this work is to provide an explicit $W$-triangulation and to describe the associated cellular homology cochain complex, as a ${\mathbb Z}[W]$-dg-algebra. This study is motivated by the research of equivariant cellular structures in Lie theory, and more precisely for flag manifolds and classifying spaces of tori.

Let $\mathfrak{t}$ denote the Lie algebra of $T$. The exponential map $\exp : \mathfrak{t} \to T$ induces a $W$-equivariant isomorphism of Lie groups $\mathfrak{t}/\Lambda \to T$, where $\Lambda\subset\mathfrak{t}$ is a $W$-lattice. The isomorphism type of the pair $(K,T)$ is determined by the root system $\Phi\subset i\mathfrak{t}^*$ and the lattice $\Lambda$. This is equivalent to the \emph{root datum} of $(K,T)$ and this gives the suitable vocabulary to work with.

An important distinction comes from the fundamental group $\pi_1(K)$ of $K$. In the case where $\pi_1(K)=1$, the combinatorics of the \emph{affine Weyl group} $W_\mathrm{a}$ and alcoves easily give the desired CW-structure. Indeed, the group $W_\mathrm{a}$ is Coxeter with one additional generator, corresponding to the reflection of the highest root of $\Phi$. Moreover, the fundamental alcove is a standard simplex and its triangulation provides a $W$-equivariant triangulation of $T$. We also give a formula for the cup product (see Corollary \ref{cupproductforT}).

In the general case, the \emph{extended affine Weyl group} $W_\Lambda:=\Lambda\rtimes W$ is no longer a Coxeter group and the above combinatorics does not hold anymore. This comes from the non-trivial symmetries of the fundamental alcove in the group. However, it is enough to consider the barycentric subdivision of the fundamental alcove (see Theorem \ref{complexforY}). Though heavy in computations, this has the advantage of giving a general statement for all cases at once. Moreover, this construction applied to the simply-connected case gives the same complex as the first one, up to ${\mathbb Z}[W]$-homotopy equivalence.

In the simply-connected case, the complex is described using minimal length coset representatives, but if we rewrite it using transitive sets, then the combinatorics makes sense for every finite group with a preferred element and set of generators. If moreover a notion of parabolic subgroups is available, then the cup product, as described in Theorem \ref{cupprodparabolics} makes sense as well. In particular, we can construct such a complex for every finite (irreducible) Coxeter group.

It is a natural question to ask if there still is any geometric information behind the above complex in the non-crystallographic Coxeter groups. Our main result is the following one, which gathers Theorem \ref{defsteinbergtorus}, Corollary \ref{T(I2(m))hyperbolicsurface} and Proposition \ref{definableoverQbar}:

\begin{theo*}
If $W$ is a finite irreducible Coxeter group of rank $n$, then there exists an orientable closed compact Riemannian $W$-manifold $\mathbf{T}(W)$ of dimension $n$, which is equivariantly diffeomorphic to a maximal torus of a simply-connected compact Lie group of the same type as $W$ if $W$ is a Weyl group, and is a hyperbolic manifold otherwise.

Moreover, if $W$ is one of the groups $I_2(2g+1)$, $I_2(4g)$ or $I_2(4g+2)$ (with $g>0$), then $\mathbf{T}(W)$ is naturally a Riemann surface of genus $g$, definable over $\overline{{\mathbb Q}}$ and an elliptic curve if $g=1$.
\end{theo*}

It should be mentioned that the manifolds $\mathbf{T}(H_3)$ and $\mathbf{T}(H_4)$ were already constructed respectively by Zimmermann (\cite{zimmermann_hyperbolic}) and by Davis (\cite{davis_hyperbolic}), using a different method.

Our construction relies on the choice of a particular reflection of $W$, which is the one associated to the highest root in the root system of $W$, in all cases except $H_3$. We use this reflection to build an infinite Coxeter group $\widehat{W}$, whose Coxeter diagram has one more node than the one of $W$ and in which $W$ is a maximal parabolic subgroup. This imitates the construction of the affine Weyl group. Furthermore, our choice implies that $\widehat{W}$ is in fact a \emph{compact hyperbolic} Coxeter group (see \cite[\S 6.8]{humphreys-reflectiongroups}).

This extension comes with a torsion-free normal subgroup $Q\unlhd\widehat{W}$ such that $\widehat{W}=Q\rtimes W$. A key fact is that the action of $Q$ (under the dual geometric representation of $W$) on the \emph{Coxeter complex} $\Sigma(\widehat{W})$ (seen as a quotient of the Tits cone) is a covering space action and we naturally define $\mathbf{T}(W):=\Sigma(\widehat{W})/Q$.

It is clear from the construction that $\mathbf{T}(W)$ is equipped with a $W$-triangulation (which yields a \emph{dessin d'enfant} when $W=I_2(m)$), whose associated $W$-dg-algebra has the same combinatorics as in the Weyl group case (see Theorem \ref{cochaincomplexforT(W)}). We use the \emph{Hopf trace formula} (see lemma \ref{hopfforW}) to describe the homology of $\mathbf{T}(W)$ as a representation of $W$. The following result summarizes \ref{H0andHn}, \ref{PD} and \ref{euler+poincare=love}:

\begin{theo*}
If $W$ is a finite Coxeter group of rank $n$ and $\mathbf{T}(W)$ is the $W$-manifold from the previous theorem, then we have
\[H_0(\mathbf{T}(W),{\mathbb Z})=\mathds{1}~~\text{and}~~H_n(\mathbf{T}(W),{\mathbb Z})=\varepsilon,\]
where $\varepsilon$ is the signature representation of $W$. 

The homology $H_*(\mathbf{T}(W),{\mathbb Z})$ is torsion-free and, in particular, the Betti numbers of $\mathbf{T}(W)$ are palindromic, meaning $b_i=b_{n-i}$ for all $i$.

Moreover, the geometric representation of $W$ is a direct summand of $H_1(\mathbf{T}(W),\Bbbk)$, where $\Bbbk$ is a splitting field for $W$ and $H_1(\mathbf{T}(W),\Bbbk)$ is irreducible if and only if $W$ is crystallographic.

Finally, if $W(q)$ (resp. $\widehat{W}(q)$) is the Poincar\'e series of $W$ (resp. of $\widehat{W}$), then
\[\chi(\mathbf{T}(W))=\left.\frac{W(q)}{\widehat{W}(q)}\right|_{q=1}.\]
\end{theo*}



Finally, a GAP4 package\footnote{\url{https://github.com/arthur-garnier/Salvetti-and-tori-complexes}} is provided to compute these complexes, along with the DeConcini-Salvetti complex of a finite Coxeter group (see \cite{salvetti}).

As an appendix, we use GAP again to determine all the possible hyperbolic extensions of the finite Weyl groups that correspond to a reflection in the finite group. We notice that all of them (except for $G_2=I_2(6)$) are non-compact and our construction doesn't apply; at least not immediately.





\part{Weyl-equivariant triangulations of tori of compact Lie groups and related $W$-dg-algebras}
\section{Prerequisites and notation}
\subsection{Root data, affine Weyl group and alcoves}
\hfill

\indent We start by briefly recalling some basic facts about \emph{root data} and why this is the suitable framework for our study. Standard references for what follows are \cite{malle-testerman} and \cite{kirillovandjunior}.
\begin{definition}[{\cite[Definition 9.10]{malle-testerman}}]\label{rootdatum}
A \emph{root datum} is a quadruple $(X,\Phi,Y,\Phi^\vee)$ where
\begin{enumerate}[label=(RD\arabic*)]
\item the elements $X$ and $Y$ are free abelian groups of finite rank, together with a perfect pairing $\left<\cdot,\cdot\right> : Y\times X \longrightarrow {\mathbb Z}$,
\item the subsets $\Phi\subset X$ and $\Phi^\vee\subset Y$ are (abstract) reduced root systems in ${\mathbb Z}\Phi\otimes_{\mathbb Z}{\mathbb R}$ and ${\mathbb Z}\Phi^\vee\otimes_{\mathbb Z}{\mathbb R}$, respectively,
\item there is a bijection $\Phi\longrightarrow\Phi^\vee$ (denoted by $\alpha\longmapsto \alpha^\vee$) such that $\left<\alpha^\vee,\alpha\right>=2$ for every $\alpha\in\Phi$,
\item the reflections $s_\alpha$ of the root system $\Phi$ and $s_{\alpha^\vee}$ of $\Phi^\vee$ are respectively given by
\[\forall x\in X,~s_\alpha(x):=x-\left<\alpha^\vee,x\right>\alpha\]
and
\[\forall y\in Y,~s_{\alpha^\vee}(y):=y-\left<y,\alpha\right>\alpha^\vee.\]
\end{enumerate}
The Weyl group $W$ of the root system $\Phi$ (which is isomorphic to the Weyl group of $\Phi^\vee$ via the map $s_\alpha\longmapsto s_{\alpha^\vee}$) is called the \emph{Weyl group of the root datum}. Moreover, we say that the root datum $(X,\Phi,Y,\Phi^\vee)$ is \emph{irreducible} if the root system $\Phi$ is.
\end{definition}
\indent For a root datum $(X,\Phi,Y,\Phi^\vee)$, we denote by $V:={\mathbb Z}\Phi\otimes_{\mathbb Z}{\mathbb R}$ the ambient space and $V^*:={\mathbb Z}\Phi^\vee\otimes_{\mathbb Z}{\mathbb R}$. As usual, we denote by $\Phi^+\subset\Phi$ a set of positive roots and by $\Pi\subset\Phi^+$ the corresponding set of simple roots. Define the fundamental weights $\varpi_\alpha\in V$, indexed by $\alpha\in\Pi$, by $\left<\beta^\vee,\varpi_\alpha\right>=\delta_{\alpha,\beta}$. Dually, the fundamental coweights are elements $\varpi_\alpha^\vee\in V^*$ such that $\left<\varpi_\alpha^\vee,\beta\right>=\delta_{\alpha,\beta}$. We also consider respectively
\[Q:={\mathbb Z}\Phi=\bigoplus_{\alpha\in\Pi}{\mathbb Z}\alpha\subset V~~\text{and}~~Q^\vee:={\mathbb Z}\Phi^\vee=\bigoplus_{\alpha\in\Pi}{\mathbb Z}\alpha^\vee\subset V^*\]
the \emph{root lattice} and the \emph{coroot lattice} of $\Phi$. Further, we have the respective \emph{weight lattice} and \emph{coweight lattice}:
\[P:=(Q^\vee)^\wedge=\{x\in V~;~\forall\alpha\in\Phi,~\left<\alpha^\vee,x\right>\in{\mathbb Z}\}=\bigoplus_{\alpha\in\Pi}{\mathbb Z} \varpi_\alpha\subset V~~\text{and}~~P^\vee:=\bigoplus_{\alpha\in\Pi}{\mathbb Z}\varpi_\alpha^\vee\subset V^*.\]
Thus, the abelian group $X$ is a $W$-lattice between $Q$ and $P$ and if we enumerate the simple roots $\Pi=\{\alpha_1,\dotsc,\alpha_n\}$, ($n=\dim(V)$) and if $C:=\left(\left<\alpha_i^\vee,\alpha_j\right>\right)_{1\le i,j\le n}$ is the Cartan matrix of $\Phi$, then we have
\[\det(C)=[P:Q]=[P^\vee:Q^\vee].\]

Then, the Chevalley classification theorem (see \cite[\S 9.2]{malle-testerman}) says that, given a connected reductive (complex) group $G$ and $T$ a maximal torus of $G$, if $\Phi$ denotes the root system of $(G,T)$, if $X(T):=\mathrm{Hom}\,(T,\mathbb{G}_m({\mathbb C}))$ and $Y(T):=\mathrm{Hom}\,(\mathbb{G}_m({\mathbb C}),T)$ are the respective character and cocharacter lattice of $T$, then $(X(T),\Phi,Y(T),\Phi^\vee)$ is a root datum that characterizes $G$ (and $T$) up to isomorphism. 

This is also true for Lie groups. Let $K$ be a simple compact Lie group, $T$ a maximal torus of $K$, we denote by $\mathfrak{k}$ and $\mathfrak{t}$ their respective Lie algebras and by $\Phi\subset i\mathfrak{t}^*=:V$ the root system of $(\mathfrak{k},\mathfrak{t})$. We may take
\[X(T)=\{d\lambda : \mathfrak{t}\to i{\mathbb R}~;~\lambda\in\mathrm{Hom}\,(T,\mathbb{S}^1)\}\subset i\mathfrak{t}^*=V\]
and $Y(T):=X(T)^\wedge\subset V^*$, so that $(X(T),\Phi,Y(T),\Phi^\vee)$ is a root datum. Since $T$ is abelian, the Weyl group $W=W(\Phi)\simeq N_K(T)/T$ acts on $T$ by conjugation by a representative in the normalizer.

By \cite[Lemma 1]{kirillovandjunior}, the normalized exponential map defines a $W$-isomorphism of Lie groups
\begin{equation}\label{T=V/Y}\tag{$\dagger$}
V^*/Y(T) \stackrel{\tiny{\sim}}\longrightarrow T.
\end{equation}
Moreover, we have the following isomorphisms
\[P/X(T)\simeq\pi_1(K)~~\text{and}~~X(T)/Q\simeq Z(K).\]

This shows that we may reformulate the initial problem as follows: given an irreducible root datum $(X,\Phi,Y,\Phi^\vee)$ with Weyl group $W$ and ambient space $V:={\mathbb Z}\Phi\otimes_{\mathbb Z}{\mathbb R}$, find a $W$-equivariant triangulation of the torus $V^*/Y$. As mentioned above, this will depend on the \emph{fundamental group} $P/X$ of the root datum.
\begin{notation*}
Throughout the first part of the paper we fix, once and for all, an irreducible root datum $(X,\Phi,Y,\Phi^\vee)$, with ambient space $V={\mathbb Z}\Phi\otimes {\mathbb R}$, simple roots $\Pi\subset\Phi^+$, Weyl group $W=\left<s_\alpha,~\alpha\in\Pi\right>$, fundamental (co)weights $(\varpi_\alpha)_{\alpha\in\Pi}$ and $(\varpi^\vee_\alpha)_{\alpha\in\Pi}$, (co)root lattices $Q$ and $Q^\vee$ and (co)weight lattices $P$ and $P^\vee$.
\end{notation*}
\indent Recall that $\Phi$ has a unique \emph{highest root}, i.e. a positive root $\alpha_0=\sum_{\alpha\in\Pi}n_{\alpha}\alpha\in\Phi^+$ such that $\alpha\le\alpha_0$ for all $\alpha\in\Phi$ (see \cite[\S 10.4, Lemma A]{humlie} or \cite[\S 11.2]{kane}). We consider the affine transformation
\[s_0:=\mathrm{t}_{\alpha_0^\vee}s_{\alpha_0} : \lambda \longmapsto s_{\alpha_0}(\lambda)+\alpha_0^\vee=\lambda-(\left<\lambda,\alpha_0\right>-1)\alpha_0^\vee.\]
Then, the group $W_\mathrm{a}:=\left<s_0,s_1,\dotsc,s_n\right>\le\mathrm{Aff}(V^*)$ is a Coxeter group, called the \emph{affine Weyl group}. It splits as
\[W_\mathrm{a}=Q^\vee\rtimes W.\]
Moreover, for $\alpha\in\Phi$ and $k\in{\mathbb Z}$, we consider the affine hyperplanes $H_{\alpha,k}:=\{\lambda\in V^*~;~\left<\lambda,\alpha\right>=k\}$ and we call \emph{alcove} any connected component of $V^*\setminus\bigcup_{\alpha,k}H_{\alpha,k}$. The \emph{fundamental alcove} is
\[\mathcal{A}_0:=\{\lambda\in V^*~;~\forall \alpha\in\Phi^+,~0<\left<\lambda,\alpha\right><1\}=\{\lambda\in V^*~;~\forall \alpha\in\Pi,~\left<\lambda,\alpha\right>>0,~\left<\lambda,\alpha_0\right><1\}.\]
Then, by \cite[V, \S 2.2, Corollaire]{bourbaki456}, its closure is a standard simplex
\[\overline{\mathcal{A}_0}=\mathrm{conv}\left(\{0\}\cup\left\{\frac{\varpi_\alpha^\vee}{n_\alpha}\right\}_{\alpha\in\Pi}\right)\simeq\Delta^n\]
and by \cite[\S 4.5 and 4.8]{humphreys-reflectiongroups}, $\overline{\mathcal{A}_0}$ is a fundamental domain for $W_\mathrm{a}$ in $V^*$ and moreover, $W_\mathrm{a}$ acts simply transitively on the set of open alcoves.





\subsection{Reminder on equivariant cellular structures and related chain complexes}
\hfill

Before going any further into our study, we shall give some reminders on equivariant CW-complexes. This will also set the notation for the sequel. The following treatment can essentially be found in \cite[II, \S 1]{tomdieck_transgroups}.

Let $X$ be a CW-complex (see \cite[Chapter 0 and Appendix]{hatcher}). For $n\in{\mathbb N}$, we denote by $\mathcal{E}_n(X)$ the set of $n$-cells of $X$. For instance, the $n$-skeleton of $X$ is given by $X_n=\bigcup_{k\le n}\bigcup_{e\in\mathcal{E}_k(X)}e$.

\begin{definition}\label{GCW}
Let $G$ be a discrete group and $X$ be a $G$-space, that is also a CW-complex (in particular, Hausdorff). We say that $X$ is a \emph{$G$-CW-complex} if the action of $G$ on $X$ induces an action of $G$ on $\mathcal{E}_n(X)$ for all $n\in{\mathbb N}$ and, for every cell $e$ of $X$ and any $g\in G$, if $ge=e$, then $gx=x$ for each $x\in e$.
\end{definition}

This is not the right definition of a $G$-CW-complex (see \cite{tomdieck_transgroups}, in particular we can remove the assumption that $X$ should be Hausdorff), but the two definitions are equivalent on a Hausdorff space as soon as the acting group $G$ is discrete, which is our case.

Given a $G$-CW-complex $X$, we may consider its \emph{integral cellular homology chain complex} $C^\mathrm{cell}_*(X,G;{\mathbb Z})$ (see \cite[\S 2.2]{hatcher}). In particular, in homogeneous degree $n$, we have
\[C^\mathrm{cell}_n(X,G;{\mathbb Z})={\mathbb Z}\left<\mathcal{E}_n(X)\right>\stackrel{\tiny{\text{df}}}=\bigoplus_{e\in\mathcal{E}_n(X)}{\mathbb Z} e.\]
and since $\mathcal{E}_n(X)$ is a $G$-set, this complex is a complex of permutation ${\mathbb Z}[G]$-modules:
\[C^\mathrm{cell}_n(X,G;{\mathbb Z})\simeq \bigoplus_{[e]\in\mathcal{E}_n(X)/G}{\mathbb Z}[G/G_e],\]
where we denote by $G_e:=\mathrm{Stab}_G(e)$ the stabilizer of the cell $e\in\mathcal{E}_n(X)$.

By \cite[II, \S 1.17, Exercise 2]{tomdieck_transgroups}, if $X$ is a $G$-CW-complex, then the orbit space $X/G$ (if it is Hausdorff) naturally inherits the structure of a classical CW-complex with $n$-skeleton $(X/G)_n=X_n/G$. We will need a slight generalization of this here.

Recall that if $G$ acts on a set $X$ and if $N\unlhd G$, then we may consider the \emph{deflation} of $X$:
\[\mathrm{Def}^G_{G/N}(X):=X/N,\]
with the induced action of $G/N$.

If $\pi : G\twoheadrightarrow G/N$ is the projection map and if $X=G/H$ is a transitive $G$-set, then we have a canonical isomorphism of $G/N$-sets
\[\mathrm{Def}^G_{G/N}(G/H)\simeq \pi(G)/\pi(H).\]
This gives a functor $G\mathbf{-Set}\to G/N\mathbf{-Set}$, whose linearization is the usual linear deflation
\[\begin{array}{ccccc}
\mathrm{Def}_{G/N}^G & : & {\mathbb Z}[G]\mathbf{-Mod} & \longrightarrow & {\mathbb Z}[G/N]\mathbf{-Mod} \\ & & U & \longmapsto & U_N:=U/\left<nu-u\right>\end{array}\]
and we may extend this functor to (co)chain complex categories.
\begin{lem}\label{GCWandSDP}
Let $G$ be a discrete group, written as a semi-direct product $G=N\rtimes H$ and $X$ be an $G$-CW-complex. Denote by $p : X\twoheadrightarrow X/N$ and by $\pi : G\twoheadrightarrow H$ the natural projections.
If the quotient space $X/N$ is Hausdorff, then it is an $H$-CW-complex such that, for all $k\in{\mathbb N}$,
\[\mathcal{E}_k(X/N)=\{p(e),~e\in\mathcal{E}_k(X)\}\]
and the map $\pi$ induces a natural isomorphism
\[C^\mathrm{cell}_*(X/N,H;{\mathbb Z})\simeq \mathrm{Def}^G_H\left(C^\mathrm{cell}_*(X,G;{\mathbb Z})\right).\]
\end{lem}
\begin{proof}
The fact that $X/N$ is a CW-complex with the given set of $k$-cells is standard and comes from the fact that $X/N$ is Hausdorff and that the usual relation $x\sim y~\Leftrightarrow~\exists g\in G~;~y=gx$ is \emph{cellular} (see \cite[Definition I.6.1 and Theorems I.6.2 and II.5.7]{lundell-weingram}). Moreover, we immediately verify that the resulting structure is $H$-cellular.

Now, the map
\[\begin{array}{ccccc}
\mathcal{E}_k(X)/N & \to & \mathcal{E}_k(X/N) & \twoheadrightarrow \mathcal{E}_k(X/N)/H \\ N\cdot e & \mapsto & p(e) & & \end{array}\]
together with the decomposition $G=N\rtimes H$ gives a natural bijection
\[\mathcal{E}_k(X)/G \stackrel{\tiny{\approx}}\longrightarrow \mathcal{E}_k(X/N)/H\]
and we have $H_{p(e)}\simeq\pi(G_e)$, yielding the desired isomorphism.
\end{proof}
\begin{rem}
Similar arguments show that we may replace deflations in the previous statement by inflation, restriction or induction. Inflation and restriction are obvious to define and, given a discrete group $G$, a subgroup $H\le G$ and an $H$-space $X$, we may define $\mathrm{Ind}_H^G(X):=G\times_H X=(G\times X)/H$, see \cite[\S 2.3]{bouc2010}. If $X$ is a $H$-CW-complex with cells $e$, then this is a $G$-CW-complex with cells $\{g\}\times_He$ for $g\in G$.
\end{rem}














\section{The simply-connected case}


\subsection{The $W_\mathrm{a}$-triangulation of $V^*$ associated to the fundamental alcove}
\hfill

The problem of finding a $W$-equivariant triangulation of $T=V^*/Y$ lifts to finding a $(Y\rtimes W)$-equivariant triangulation of $V^*$. In the 1-connected case, we have $Y=Q^\vee$ and $Q^\vee\rtimes W=W_\mathrm{a}$.
 
As the alcove $\overline{\mathcal{A}_0}$ is a fundamental domain for $W_\mathrm{a}$ acting on $V^*$, it suffices to have an equivariant triangulation of $\overline{\mathcal{A}_0}$. The fundamental result is the Theorem (or more precisely, its proof) from \cite[\S 4.8]{humphreys-reflectiongroups}, which ensures that the natural polytopal structure on the $r$-simplex $\overline{\mathcal{A}_0}$ is $W_\mathrm{a}$-equivariant.

\iffalse
\begin{lem}[{\cite[\S 4.4 and \S 4.5]{humphreys-reflectiongroups}}]\label{y=wxyieldsy=x}
Let $\lambda,\lambda'\in\overline{\mathcal{A}_0}$ and $\widetilde{w}\in W_\mathrm{a}$ such that $\lambda'=\widetilde{w}\lambda$. Then $\lambda=\lambda'$ and $\widetilde{w}$ is a product of simple reflections in $S_0$, each one of which fixes $\lambda$.
\newline
In particular, two $k$-faces are in the same $W_\mathrm{a}$-orbit if and only if they are equal.
\end{lem}
\fi

Now we introduce some notation about faces of polytopes.

Given a polytope $\mathcal{P}\subset{\mathbb R}^n$ and an integer $k\ge -1$, we denote by $F_k(\mathcal{P})$ the set of $k$-dimensional faces of $\mathcal{P}$. In particular, we have $F_{-1}(\mathcal{P})=\{\emptyset\}$, $F_{\dim(\mathcal{P})}(\mathcal{P})=\{\mathcal{P}\}$, $F_{\dim(\mathcal{P})-1}(\mathcal{P})$ is the set of facets of $\mathcal{P}$ and $F_0(\mathcal{P})=\vertices(\mathcal{P})$ is its set of vertices. Moreover, we let $F(\mathcal{P}):=\bigcup_kF_k(\mathcal{P})$ be the \emph{face lattice} of $\mathcal{P}$. It is indeed a lattice for the inclusion relation.

Notice that the above vocabulary also applies to a polytopal complexes, that is, a finite family of polytopes glued together along common faces.

Resuming to root data, we consider the hyperplanes
\[\forall i\in S:=\{1,\dotsc,n\},~H_i:=H_{\alpha_i,0}=\{\lambda\in V^*~;~\left<\lambda,\alpha_i\right>=0\}\]
and
\[H_0:=H_{\alpha_0,1}=\{\lambda\in V^*~;~\left<\lambda,\alpha_0\right>=1\}\]
with $\alpha_0=\sum_in_i\alpha_i$ the highest root. We also take the following notation for the vertices of $\overline{\mathcal{A}_0}$:
\[\forall i\in S,~v_i:=\frac{\varpi_i^\vee}{n_i}~~\text{and}~~v_0:=0~~\text{so that}~~\vertices(\overline{\mathcal{A}_0})=\{v_0,v_1,\dotsc,v_n\}.\]
The hyperplane $H_i$ for $i\in S_0:=S\cup\{0\}$ give a complete set of bounding hyperplanes for the $n$-simplex $\overline{\mathcal{A}_0}$. Furthermore, by definition, for every face $f\in F_k(\overline{\mathcal{A}_0})$ there exists a subset $I\subseteq S_0$ of cardinality $|I|=\mathrm{codim}\,_{\overline{\mathcal{A}_0}}(f)=n-k$ such that 
\[f=f_I:=\overline{\mathcal{A}_0}\cap\bigcap_{i\in I}H_i\]
and we readily have
\[\vertices(f_I)=\{v_i~;~i\in S_0\setminus I\}.\]

Recall also that for $i\in S_0$, denoting by $s_i$ the reflection with respect to the hyperplane $H_i$, we have $s_i=s_{\alpha_i}\in W$ for $i\ge 1$ and $s_0=\mathrm{t}_{\alpha_0}s_{\alpha_0}$ and the group $W_\mathrm{a}$ is Coxeter, with generating system $\{s_0,s_1,\dotsc,s_n\}$. For $I\subseteq S_0$, we may consider the \emph{(standard) parabolic subgroup} $(W_\mathrm{a})_I$ of $W_\mathrm{a}$ generated by the subset $\{s_i,~i\in I\}$. If $0\notin I$, then $(W_\mathrm{a})_I)$ is in fact a parabolic subgroup of $W$.
\begin{lem}\label{stabfaceinA0}
Let $0\le k \le n$ and $I\subseteq S_0$ with $|I|=n-k$. Then the stabilizer of the face $f_I\in F_k(\overline{\mathcal{A}_0})$ is the parabolic subgroup of $W_\mathrm{a}$ associated to $I$. In other words,
\[(W_\mathrm{a})_{f_I}=(W_\mathrm{a})_I.\]
\end{lem}
\begin{proof}
As $\vertices(f_I)=\{v_i,~i\notin I\}$ is $(W_\mathrm{a})_{f_I}$-stable, the Theorem from \cite[\S 4.8]{humphreys-reflectiongroups} ensures that
\[(W_\mathrm{a})_{f_I}=\bigcap_{i\in S_0\setminus I}(W_\mathrm{a})_{v_i}.\]
Moreover, each group $(W_\mathrm{a})_{v_j}$ is generated by the reflections it contains, so that $v_j\in H_i$. A reflection $s_i$ fixes $0$ if and only if it is linear, so $(W_\mathrm{a})_{v_0}=\left<s_i~;~i\ne0\right>=(W_\mathrm{a})_S=W$. Let now $j\in S$. Since 
\[\{v_j\}=\left\{\frac{\varpi_j^\vee}{n_j}\right\}=\bigcap_{j\ne i\in S_0}H_i,\]
we have that $s_i(v_j)=v_j$ if and only if $i\in S_0\setminus\{j\}$ and hence, for every $j\in S_0$, we have
\[(W_\mathrm{a})_{v_j}=(W_\mathrm{a})_{S_0\setminus\{j\}}\]
and thus
\[(W_\mathrm{a})_{f_I}=\bigcap_{i\in S_0\setminus I}(W_\mathrm{a})_{v_i}=\bigcap_{i\in S_0\setminus I}(W_\mathrm{a})_{S_0\setminus\{i\}}=(W_\mathrm{a})_{\bigcap_{i\notin I}S_0\setminus\{i\}}=(W_\mathrm{a})_I.\]
\end{proof}

Hence, we have a triangulation
\[V^*=\coprod_{\substack{\widetilde{w}\in W_\mathrm{a} \\ f\in F(\overline{\mathcal{A}_0})}}\widetilde{w}\cdot f\]
which is $W_\mathrm{a}$-equivariant and following the notation from the beginning of this section, we have $\mathcal{E}_k(V^*)/W_\mathrm{a}=F_k(\overline{\mathcal{A}_0})$ for all $k$. Therefore, we get isomorphisms of ${\mathbb Z}[W_\mathrm{a}]$-modules
\[C^\mathrm{cell}_k(V^*,W_\mathrm{a};{\mathbb Z})\simeq\bigoplus_{f\in F_k(\overline{\mathcal{A}_0})}{\mathbb Z}[W_\mathrm{a}/(W_\mathrm{a})_f]=\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}[W_\mathrm{a}/(W_\mathrm{a})_I].\]

We have to fix an orientation of the cells in $V^*$ and determine their boundary. But each one of them is a simplex, so its orientation is determined by an orientation on its vertices. We choose to orient them as the index set $(S_0,\le)$. For $I\subseteq S_0$ with corresponding $k$-face $f_I=\conv(\{v_i~;~i\in S_0\setminus I\})$, we write
\[f_I=[v_{j_1},\dotsc,v_{j_{k+1}}]~~\text{with}~~\{j_1<j_2<\dotsc<j_{k+1}\}=S_0\setminus I\]
to make its orientation explicit. The oriented boundary of $f_I$ is then simply given by the formula
\[\partial_k(f_I)=\sum_{u=1}^{k+1}(-1)^u\underbrace{[v_{j_1},\dotsc,\widehat{v_{j_u}},\dotsc,v_{j_{k+1}}]}_{=\conv(\{v_j~;~j_u\ne j\in S_0\setminus I\})}=\sum_{u=1}^{k+1}(-1)^uf_{I\cup\{j_u\}}\]

We have thus obtained the following result:
\begin{theo}\label{complexforWa}
The face lattice of the $n$-simplex $\overline{\mathcal{A}_0}$ induces a $W_\mathrm{a}$-equivariant triangulation of $V^*$, whose cellular complex $C^\mathrm{cell}_*(V^*,W_\mathrm{a};{\mathbb Z})$ is given (in homogeneous degrees $k$ and $k-1$) by
\[\xymatrix{\cdots \ar[r] & \displaystyle{\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}\left[W_\mathrm{a}^I\right]} \ar^<<<<<{\partial_k}[r] & \displaystyle{\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k+1}}{\mathbb Z}\left[W_\mathrm{a}^I\right]} \ar[r] & \cdots}\]
where $W_\mathrm{a}^I\approx W_\mathrm{a}/(W_\mathrm{a})_I$ is the $W_\mathrm{a}$-set of minimal length left coset representatives, modulo the parabolic subgroup $(W_\mathrm{a})_I$ and boundaries are defined as follows:
\[\forall k\in{\mathbb N},~\forall I\subset S_0~;~S_0\setminus I=\{j_1<\dotsc <j_{k+1}\},~{(\partial_k)}_{\mid{\mathbb Z}\left[W_\mathrm{a}^I\right]}=\sum_{u=1}^{k+1}(-1)^u p^I_{I\cup\{j_u\}},\]
where, for $I\subset J$, $p^{I}_J$ denotes the projection
\[p^{I}_J : W_\mathrm{a}^{I}=W_\mathrm{a}/(W_\mathrm{a})_{I} \longtwoheadrightarrow W_\mathrm{a}/(W_\mathrm{a})_J=W_\mathrm{a}^J.\]
\end{theo}








\subsection{The $W$-dg-algebra structure}
\hfill

We now make the cup product on $C^\mathrm{cell}_*(V^*,W_\mathrm{a};{\mathbb Z})$ more explicit. We first notice that, for a finite group $G$ and $H\le G$, we have a canonical isomorphism of right ${\mathbb Z}[G]$-modules
\[\begin{array}{ccc}
{\mathbb Z}[G/H]^\vee\stackrel{\tiny{\text{df}}}=\mathrm{Hom}\,({\mathbb Z}[G/H],{\mathbb Z}) & \longrightarrow & {\mathbb Z}[H\backslash G] \\ (gH)^* & \longmapsto & Hg^{-1}\end{array}\]
Moreover, for a cell $e\in C^\mathrm{cell}_*(V^*,W_\mathrm{a};{\mathbb Z})$, we denote by $e^*\in C_\mathrm{cell}^*(V^*,W_\mathrm{a};{\mathbb Z})=C_*^\mathrm{cell}(V^*,W_\mathrm{a};{\mathbb Z})^\vee$ its dual.

Finally recall that the cup product of two (dual) simplices is given by the formula
\[[u_0,\dotsc,u_k]^*\cup [v_0,\dotsc,v_l]^*=\delta_{u_k,v_0}[u_0,\dotsc,u_k,v_1,\dotsc,v_l]^*.\]

We may express this product on $C^\mathrm{cell}_*(V^*,W_\mathrm{a};{\mathbb Z})$ in terms of parabolic double cosets. We write
\[C_\mathrm{cell}^k(V^*,W_\mathrm{a};{\mathbb Z})=\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}\left[W_\mathrm{a}/(W_\mathrm{a})_I\right]^\vee=\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}\left[{}^I{W_\mathrm{a}}\right],\]
where 
\[{}^I{W_\mathrm{a}}\stackrel{\tiny{\text{df}}}=\{w\in W_\mathrm{a}~;~\ell(s_iw)>\ell(w),~\forall i\in I\}\approx (W_\mathrm{a})_I\backslash W_\mathrm{a}\]
is the set of minimal length right coset representatives. Recall the following general result about double cosets:
\begin{lem}[{\cite[\S 3, Proposition 2 and Corollary 3]{billey2016}}]\label{DCD}
Let $(W,S)$ be a Coxeter system and $I,J\subset S$. Denote as usual
\[W^I:=\{w\in W~;~\ell(ws)>\ell(w),~\forall s\in I\}\approx W/W_I,\]
\[{}^IW:=\{w\in W~;~\ell(sw)>\ell(w),~\forall s\in I\}\approx W_I\backslash W\]
and
\[J\subset I~\Rightarrow~W^J_I:=\{w\in W_I~;~\ell(ws)>\ell(w),~\forall s\in J\}\approx W_I/W_J.\]
\begin{enumerate}
\item Each double coset in $W_I\backslash W/W_J$ has a unique element of minimal length.
\item An element $w\in W$ is of minimal length in its double coset if and only if $w\in {}^IW\cap W^J$. In particular, we have a bijection 
\[W_I\backslash W/W_J\approx {}^IW\cap W^J.\]
\item As a consequence, if $w\in {}^IW\cap W^J$ and $x\in W_I$, then $xw\in W^J$ if and only if $x\in W^{I\cap {}^wJ}_I$. Hence, we have the following property:
\[\forall x\in W_IwW_J,~\exists!(u,v)\in W^{I\cap{}^wJ}_I\times W_J~;~\left\{\begin{array}{ll}x=uwv, \\ 
\ell(x)=\ell(u)+\ell(w)+\ell(v). \end{array}\right.\]
\end{enumerate}
\end{lem}
\bigskip

We can now formulate the main result:
\medskip
\begin{theo}\label{cupprodparabolics}
The ${\mathbb Z}[W_\mathrm{a}]$-cochain complex $C_\mathrm{cell}^*(V^*,W_\mathrm{a};{\mathbb Z})$ associated to the $W_\mathrm{a}$-triangulation of $V^*$ is a ${\mathbb Z}[W_\mathrm{a}]$-dg-algebra with homogenous components
\[\forall 0\le k \le n,~C^k_\mathrm{cell}(V^*,W_\mathrm{a};{\mathbb Z})=\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}\left[(W_\mathrm{a})_I\backslash W_\mathrm{a}\right]\simeq\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}\left[{}^I{W_\mathrm{a}}\right]\]
and differentials
\[\forall I\subset S_0~;~S_0\setminus I=:\{j_0<\cdots<j_k\},~d^k({}^I{w})=\sum_{\substack{0\le u \le k+1 \\ j_{u-1}<j<j_u}}(-1)^u\epsilon^{I\setminus\{j\}}_Iw\]
where, by convention, $j_{-1}=-1$, $j_{k+1}=n+1$ and
\[\forall J\subset I,~{}^J_I{W_\mathrm{a}}:=\{w\in (W_\mathrm{a})_I~;~\ell(s_jw)>\ell(w),~\forall j\in J\}~~\text{and}~~{\epsilon}^J_I:=\sum_{x\in{}^J_I{W_\mathrm{a}}}x\in{\mathbb Z}\left[{}^J{W_\mathrm{a}}\right].\]

The product
\[C^p_{\mathrm{cell}}(V^*,W_\mathrm{a};{\mathbb Z}) \otimes C^q_{\mathrm{cell}}(V^*,W_\mathrm{a};{\mathbb Z}) \longrightarrow C^{p+q}_{\mathrm{cell}}(V^*,W_\mathrm{a};{\mathbb Z})\]
is induced by the cup product
\[{\mathbb Z}\left[{}^I{W_\mathrm{a}}\right] \otimes {\mathbb Z}\left[{}^J{W_\mathrm{a}}\right] \stackrel{\tiny{\cup}}\longrightarrow {\mathbb Z}\left[{}^{I\cap J}{W_\mathrm{a}}\right]\]
defined by the formula
\[{}^Ix\cup{}^Jy=\delta_{\max(I^\complement),\min(J^\complement)}\times\left\{\begin{array}{cc}
{}^{I\cap J}{((xy^{-1})_Jy)} & \text{if }xy^{-1}\in (W_\mathrm{a})_I(W_\mathrm{a})_J, \\ 0 & \text{otherwise},\end{array}\right.\]
where, given $w\in W_\mathrm{a}$, we denote by ${}^{I\cap J}w\in{}^{I\cap J}{W_\mathrm{a}}$ its minimal length right coset representative and if $(W_\mathrm{a})_Iw(W_\mathrm{a})_J=(W_\mathrm{a})_I(W_\mathrm{a})_J$, $w_J$ is the unique element $v\in (W_\mathrm{a})_J$ such that $w=uv$, with $u\in (W_\mathrm{a})_I^{I\cap J}$ and $\ell(w)=\ell(u)+\ell(v)$.
\end{theo}
\medskip
\begin{proof}
Take a $k$-simplex $\sigma=[j_0,\dotsc,j_k]\subset\overline{\mathcal{A}_0}$ with $j_u\in S_0$ and set $j_{-1}:=-1$ and $j_{k+1}:=n+1$. By definition of the cochain differential $d^k$, we have
\[d^k(\sigma^*)_{|\overline{\mathcal{A}_0}}=\sum_{u=0}^{k+1}\sum_{j_{u-1}<j<j_u}(-1)^u[j_0,\dotsc,j_{u-1},j,j_u,\dotsc,j_k]^*.\]
Letting $I:=S_0\setminus\{j_0,\dotsc,j_k\}$, we have $(W_\mathrm{a})_{\sigma^*}=(W_\mathrm{a})_I$ and the above formula reads
\[d^k((W_\mathrm{a})_I\cdot1)_{|\overline{\mathcal{A}_0}}=\sum_{u=0}^{k+1}\sum_{j_{u-1}<j<j_u}(-1)^u((W_\mathrm{a})_{I\setminus\{j\}}\cdot1).\]
Therefore, as $\overline{\mathcal{A}_0}$ is a fundamental domain for $W_\mathrm{a}$ in $V^*$, this yields
\[d^k((W_\mathrm{a})_I\cdot1)=\sum_{\substack{0\le u \le n+1-|I| \\ j_{u-1}<j<j_u}}\left(\sum_{w\in {}^{I\setminus\{j\}}_I{W_\mathrm{a}}}(-1)^u((W_\mathrm{a})_{I\setminus\{j\}}\cdot w)\right),\]
which leads to the stated formula.
\smallskip

To compute the cup product, using the bijection ${}^IW_\mathrm{a}\approx (W_\mathrm{a})_I\backslash W_\mathrm{a}$, the stated formula is
\[(W_\mathrm{a})_Ix\cup(W_\mathrm{a})_Jy=\delta_{\max I^\complement,\min J^\complement}\delta_{(W_\mathrm{a})_Ixy^{-1}(W_\mathrm{a})_J,(W_\mathrm{a})_I(W_\mathrm{a})_J}(W_\mathrm{a})_{I\cap J}((xy^{-1})_Jy).\]
Let $x,y\in W_\mathrm{a}$. As $W_\mathrm{a}$ acts simplicially on $\overline{\mathcal{A}_0}$, we have 
\[(W_\mathrm{a})_Ix\cup(W_\mathrm{a})_Jy=((W_\mathrm{a})_Ixy^{-1}\cup(W_\mathrm{a})_J)y,\]
hence we may assume that $y=1$ and we just have to compute $(W_\mathrm{a})_Iw\cup(W_\mathrm{a})_J$.

First, we compute $\sigma^*\cup\tau^*$ for $\sigma,\tau\subset\overline{\mathcal{A}_0}$. As $\overline{\mathcal{A}_0}\simeq\Delta^r$ is a simplex, we may write $\sigma=[i_0,\dotsc,i_a]$ with $a=\dim\sigma$ and $I^\complement:=\{i_0,\dotsc,i_a\}\subset\mathrm{vert}(\overline{\mathcal{A}_0})\simeq S_0$. Write similarly $\tau=[j_0,\dotsc,j_b]$. We have $(W_\mathrm{a})_\sigma=(W_\mathrm{a})_{S_0\setminus \{i_0,\dotsc,i_a\}}=(W_\mathrm{a})_I$, $(W_\mathrm{a})_\tau=(W_\mathrm{a})_J$ and
\[\sigma^*\cup\tau^*=\delta_{i_a,j_0}[i_0,\dotsc,i_a,j_1,\dotsc,j_b]^*\]
and the stabilizer in $W_\mathrm{a}$ of this last dual cell is $(W_\mathrm{a})_{S_0\setminus\{i_0,\dotsc,i_a,j_1,\dotsc,j_b\}}=(W_\mathrm{a})_{I\cap J}$. Moreover, if $\sigma^*\cup\tau^*\ne0$ then we must have $i_a=j_0$, that is $\max(I^\complement)=\min(J^\complement)$. We make this assumption for the rest of this proof and we have indeed
\[\sigma^*\cup\tau^*=(W_\mathrm{a})_I\cup(W_\mathrm{a})_J=(W_\mathrm{a})_{I\cap J}.\]

\underline{Claim:} For $\tau\subset\overline{\mathcal{A}_0}$ is a simplex and $P\in F_k(V^*)$ a $k$-cell of $V^*$, if $\tau\subset P$ then $P\in (W_\mathrm{a})_\tau\cdot F_k(\overline{\mathcal{A}_0})$.

Indeed, we may assume that $\dim P=n=\dim\mathcal{A}_0$ so that there is some $z\in W_\mathrm{a}$ such that $P=z(\overline{\mathcal{A}_0})$ and so $\tau\subset\overline{\mathcal{A}_0}\cap z(\overline{\mathcal{A}_0})$, thus $z\in (W_\mathrm{a})_\tau$ (see \cite[\S 4.8]{humphreys-reflectiongroups}).
\smallskip

We are left to compute $\sigma^*w\cup\tau^*$ for $w\in W_\mathrm{a}$. If $\sigma^*w\cup\tau^*\ne0$, then $w^{-1}\sigma$ and $\tau$ are included is some common simplex $P\in{F}_k(V^*)$ and by the claim we may choose $w_\tau\in (W_\mathrm{a})_\tau=(W_\mathrm{a})_J$ such that $w^{-1}\sigma\subset w_\tau(\overline{\mathcal{A}_0})$. But then $\sigma\subset \overline{\mathcal{A}_0}\cap ww_\tau(\overline{\mathcal{A}_0})$ and so $w_\tau\sigma=w^{-1}\sigma$. This yields
\[\sigma^*w\cup\tau^*=\sigma^*w_\tau^{-1}\cup\tau^*=\sigma^*w_\tau^{-1}\cup\tau^*w_\tau^{-1}=(\sigma^*\cup\tau^*)w_\tau^{-1}=(W_\mathrm{a})_{I\cap J}\cdot w_\tau^{-1}.\]

Furthermore, if $\sigma^*w\cup\tau^*\ne0$ then we must have $ww_\tau\in (W_\mathrm{a})_I$, so $w\in(W_\mathrm{a})_I(W_\mathrm{a})_J$. In this case, the parabolic double coset decomposition from Lemma \ref{DCD} applied to the trivial double coset $(W_\mathrm{a})_Iw(W_\mathrm{a})_J$ allows one to write uniquely $w$ as $w=uw_J$ with $u\in(W_\mathrm{a})_I^{I\cap J}$ and $w_J\in(W_\mathrm{a})_J$ such that $\ell(w)=\ell(u)+\ell(w_J)$. We obtain $w_Jw_\tau\in(W_\mathrm{a})_J$ as well as $w_Jw_\tau=u^{-1}ww_\tau\in (W_\mathrm{a})_I$. Hence $w_Jw_\tau\in(W_\mathrm{a})_I\cap(W_\mathrm{a})_J=(W_\mathrm{a})_{I\cap J}$ and
\[\sigma^*w\cup\tau^*=(W_\mathrm{a})_{I\cap J}\cdot w_\tau^{-1}=(W_\mathrm{a})_{I\cap J}\cdot w_J.\]
\end{proof}
\begin{cor}\label{cupproductforT}
The ${\mathbb Z}[W]$-cochain complex $C^*_{\mathrm{cell}}(T,W;{\mathbb Z})$ associated to the $W$-triangulation of $T=V^*/Q^\vee$ induced by the $W_\mathrm{a}$-triangulation of $V^*$ is given by
\[C^*_\mathrm{cell}(T,W;{\mathbb Z})=\mathrm{Def}^{W_\mathrm{a}}_W\left(C^*_\mathrm{cell}(V^*,W_\mathrm{a};{\mathbb Z})\right).\]
In other words, if $\pi : W_\mathrm{a} \twoheadrightarrow W$ is the projection, then
\[C^k_\mathrm{cell}(T,W;{\mathbb Z})=\bigoplus_{\substack{I\subset S_0 \\ |I|=n-k}}{\mathbb Z}\left[\pi\left({}^I{W_\mathrm{a}}\right)\right]\simeq\bigoplus_{\substack{I\subset S_0 \\|I|=n-k}}{\mathbb Z}\left[\pi((W_\mathrm{a})_I)\backslash W\right],\]
with differentials
\[\forall I\subset S_0~;~S_0\setminus I=:\{j_0<\cdots<j_k\},~d^k(\pi({}^I{w}))=\sum_{\substack{0\le u \le k+1 \\ j_{u-1}<j<j_u}}(-1)^u\pi\left(\epsilon^{I\setminus\{j\}}_Iw\right),~\epsilon_I^J=\sum_{x\in{}^J_I{W_\mathrm{a}}}x\]
and product induced by the formula
\[\pi({}^Ix)\cup\pi({}^Jy)=\delta_{\max(I^\complement),\min(J^\complement)}\times\left\{\begin{array}{cc}
\pi\left({}^{I\cap J}((xy^{-1})_Jy)\right) & \text{if }xy^{-1}\in(W_\mathrm{a})_I(W_\mathrm{a})_J \\ 0 & \text{otherwise}.\end{array}\right.\]

In particular, the associated graded cohomology algebra is an integral form of the exterior algebra on $V^*$:
\[H^\bullet(C^*_\mathrm{cell}(T,W;{\mathbb R}))=H^\bullet(T,{\mathbb R})=\Lambda^\bullet(V^*).\]
\end{cor}








\section{The general case}

\subsection{The fundamental group as symmetries of an alcove}
\hfill

The group $\widehat{W_\mathrm{a}}$ acts on alcoves (transitively since $W_\mathrm{a}\unlhd\widehat{W_\mathrm{a}}$ does) but not simply-transitively. We introduce the stabilizer
\[\Omega:=\{\widehat{w}\in\widehat{W_\mathrm{a}}~;~\widehat{w}(\mathcal{A}_0)=\mathcal{A}_0\}\]
and we see that we have a decomposition $\widehat{W_\mathrm{a}}\simeq W_\mathrm{a}\rtimes\Omega$ and in particular,
\[\Omega\simeq \widehat{W_\mathrm{a}}/W_\mathrm{a}\simeq P^\vee/Q^\vee\simeq P/Q.\]
Thus, $\Omega$ is a finite abelian group. The following table details the fundamental groups of the irreducible root systems:
\begin{center}\label{fundgroups}
\begin{tabular}{|c|c|}
\hline
Type & $\Omega\simeq P/Q$ \\
\hline
\hline
$A_n~(n\ge1)$ & ${\mathbb Z}/(n+1){\mathbb Z}$ \\
\hline
$B_n~(n\ge2)$ & ${\mathbb Z}/2{\mathbb Z}$ \\
\hline
$C_n~(n\ge3)$ & ${\mathbb Z}/2{\mathbb Z}$ \\
\hline
$D_{2n}~(n\ge2)$ & ${\mathbb Z}/2{\mathbb Z} \oplus {\mathbb Z}/2{\mathbb Z}$ \\
\hline
$D_{2n+1}~(n\ge2)$ & ${\mathbb Z}/4{\mathbb Z}$ \\
\hline
$E_6$ & ${\mathbb Z}/3{\mathbb Z}$ \\
\hline
$E_7$ & ${\mathbb Z}/2{\mathbb Z}$ \\
\hline
$E_8$ & $1$ \\
\hline
$F_4$ & $1$ \\
\hline
$G_2$ & $1$ \\
\hline
\end{tabular}
\captionof{table}{Fundamental groups of irreducible root systems}
\end{center}

The description of $\Omega$ given in \cite[VI, \S 2.3]{bourbaki456} is useful. Given the highest root $\alpha_0=\sum_{i=1}^n n_i\alpha_i$ of $\Phi$, recall that a weight $\varpi_i$ is called \emph{minuscule} if $n_i=1$ and that minuscule weights form a set of representatives of the classes in $P/Q$ (see \cite[Chapter VI, Exercise 24]{bourbaki456}). Dually, we have the same notion and result for \emph{minuscule coweights}. Let 
\[M:=\{i \in S~;~n_i=1\}.\]
\begin{prop-def}\emph{(\cite[VI, \S 2.3, Proposition 6]{bourbaki456})}\label{descriptionOmega}
\newline
Let $\alpha_0=\sum_{i\in S} n_i\alpha_i$ be the highest root of $\Phi$ and $w_0\in W$ be the longest element. For $i\in S$, denote by $W_i\le W$ the Weyl group of the subsystem of $\Phi$ generated by $\{\alpha_j~;~j\ne i\}\subset\Pi$. For $i\in M$, let $w^i_0\in W_i$ be the longest element of $W_i$ and $w_i:=w^i_0w_0$. 
\newline
Then the element $\mathrm{t}_{\varpi_i^\vee}w_i\in\widehat{W_\mathrm{a}}$ is in $\Omega$ and the map
\[\begin{array}{ccc}
M & \longrightarrow & \Omega\setminus\{1\} \\ i & \longmapsto & \omega_i:=\mathrm{t}_{\varpi_i^\vee}w_i \end{array}\]
is a bijection.
\end{prop-def}

\indent We now have to see what happens if the $W$-lattice $Y$ is such that $Q^\vee \subsetneq Y \subsetneq P^\vee$. To simplify notation of this section, we identify a lattice $\Lambda\subset V^*$ with its translation group $\mathrm{t}(\Lambda)\subset \mathrm{Aff}(V^*)$. We define the \emph{intermediate affine Weyl group} $W_Y:=\mathrm{t}(Y)\rtimes W$.
\newline
\indent First, we shall identify $Y$ with a subgroup of $\Omega$. In fact, there is a correspondence between $W$-lattices $Q^\vee\subseteq\Lambda\subseteq P^\vee$ and the subgroups of $\Omega$. In order to state this correspondence properly, we temporarily drop the letter $Y$ and we work in the root system $\Phi$ only. Though straightforward, the following result is key:
\begin{prop}\label{identlatticessubgroupsofOmega}
Recall that $\widehat{W_\mathrm{a}}\simeq W_\mathrm{a}\rtimes\Omega$ and denote by 
\[\pi : \widehat{W_\mathrm{a}}\longtwoheadrightarrow \Omega\]
the natural projection. Given a $W$-lattice $Q^\vee\subset\Lambda\subset P^\vee$, we define a subgroup $W_\Lambda:=\Lambda\rtimes W\le\widehat{W_\mathrm{a}}$. Then we have a bijective correspondence
\[\begin{array}{ccc}
\left\{\Lambda~;~Q^\vee\subseteq\Lambda\subseteq P^\vee~\text{is a } W\text{-lattice}\right\} & \stackrel{\tiny{\mathrm{1-1}}}\longleftrightarrow & \left\{H\le \Omega\right\} \\
\Lambda & \longmapsto & \Omega_\Lambda:=\pi(W_\Lambda) \\ \pi^{-1}(H)\cap P^\vee=:\Lambda(H) & \longmapsfrom & H \end{array}\]
Moreover, for a $W$-lattice $Q^\vee \subset \Lambda\subset P^\vee$, we have
\[[\Omega:\Omega_\Lambda]=[P^\vee:\Lambda],~\text{equivalently},~|\Omega_\Lambda|=[\Lambda:Q^\vee].\]
Finally, we have a decomposition
\[W_\Lambda\simeq W_\mathrm{a}\rtimes \Omega_\Lambda.\]
\end{prop}

\iffalse
\begin{proof}
It is clear that the map $\Omega_\bullet$ is well-defined. It is also the case for $\Lambda(\bullet)$. Indeed, if $H\le\Omega$, then $\Lambda(H)=\pi^{-1}(H)\cap P^\vee$ is a subgroup of $W_\mathrm{a}$ and it is discrete since it is a subset of $P^\vee$. We also have $Q^\vee\subset\Lambda(H)\subset P^\vee$, so $\mathrm{rk}(\Lambda(H))=\mathrm{rk}(Q)=r$. Moreover, it is clear that $\Lambda(H)$ is $W$-stable.

Next, it is routine to verify that $\Lambda(\bullet)$ and $\Omega_\bullet$ are bijective and inverse to each other.

Finally, the fact that $[P^\vee:\Lambda]\cdot [\Lambda:Q^\vee]=[P^\vee:Q^\vee]=|\Omega|$ ensures that the two equalities are indeed equivalent. But since $\ker(\pi)=W_\mathrm{a}\le W_\Lambda$ we have
\[\Omega_\Lambda=\pi(W_\Lambda)=\mathrm{im}\,(W_\Lambda \stackrel{\tiny{\pi}}\to\Omega)\simeq W_\Lambda/\ker(\pi_{|W_\Lambda})=W_\Lambda/\ker(\pi)=W_\Lambda/W_\mathrm{a}\simeq \Lambda/Q^\vee.\]

Finally, using the above equality we see that the short exact sequence
\[\xymatrix{1 \ar[r] & W_\mathrm{a} \ar[r] & W_\Lambda \ar[r] & \Omega_\Lambda \ar[r] & 1}\]
is exact. It splits since the following sequence is split:
\[\xymatrix{1 \ar[r] & W_\mathrm{a} \ar[r] & \widehat{W_\mathrm{a}} \ar^\pi[r] & \Omega \ar[r] & 1}\]
\end{proof}
\fi

\iffalse
\begin{proof}
First, we prove that the maps $\Omega_\bullet$ and $\Lambda(\bullet)$ are well-defined. It is clear that, given a $W$-lattice $\Lambda$, the set $\Omega_\Lambda=\pi(\Lambda\rtimes W)=\{\phi\in W_\Lambda~;~\phi(\mathcal{A}_0)=\mathcal{A}_0\}$ is a subgroup of $\Omega$. Conversely, le $H\le\Omega$ be a subgroup. We have that $\Lambda(H)=\pi^{-1}(H)\cap P^\vee$ is a subgroup of $\widehat{W_\mathrm{a}}$ and since $\Lambda(H)\subset P^\vee$, $\Lambda(H)$ is countable, hence discrete. We readily have $Q^\vee\subset\Lambda(H)\subset P^\vee$, which gives $\mathrm{rk}\,(\Lambda(H))=\mathrm{rk}\,(Q)=\mathrm{rk}\,(\Phi)$. 

Moreover, if $x\in\Lambda(H)$ and $w\in W$, then we have $\pi(x)=\pi(x\cdot\mathrm{t}_0w)=\pi(\mathrm{t}_0w\cdot x)$ (recall that $\Omega$ is abelian) so $wx\in\pi^{-1}(H)$ that is, $wx\in\Lambda(H)$ and $\Lambda(H)$ is indeed a $W$-lattice lying between $Q^\vee$ and $P^\vee$.

We have to prove that $\Lambda\circ \Omega_\bullet=id$ and $\Omega_{\Lambda(\bullet)}=id$. Take a $W$-lattice $\Lambda$ between $Q^\vee$ and $P^\vee$ and let us show that $\Lambda(\Omega_\Lambda)=\Lambda$. If $\mu\in \Lambda$, then $\mu\in P^\vee$ and by construction, $\pi(\mathrm{t}_\mu)\in\pi(\Lambda\rtimes W)$ so $\mu\in \pi^{-1}(\pi(W_\Lambda))\cap P^\vee\stackrel{\tiny{\text{def}}}=\Lambda(\Omega_\Lambda)$. On the other hand, if $\mu\in \pi^{-1}(\pi(W_\Lambda))\cap P^\vee$, then there are $\nu\in \Lambda$ and $w\in W$ such that $\pi(\mathrm{t}_\mu)=\pi(\mathrm{t}_\nu w)$, that is, $1=\pi(\mathrm{t}_\nu w \mathrm{t}_{-\mu})=\pi(\mathrm{t}_{\nu-w(\mu)}w)$. Hence $\mathrm{t}_{\nu-w(\mu)}w\in W_\mathrm{a}$, which means that $\nu-w(\mu)\in Q^\vee\subset\Lambda$, but since $\nu\in\Lambda$ and $\Lambda$ is a $W$-lattice, we have $\mu\in\Lambda$.

Now, let $H\le\Omega$ be a subgroup and let us prove that $\Omega_{\Lambda(H)}=H$. Let $x\in \Omega_{\Lambda(H)}=\pi(W_{\Lambda(H)})$. There exist $\mathrm{t}_\mu w\in \Lambda(H)\rtimes W$ such that 
\[x=\pi(\mathrm{t}_\mu w)=\pi(\mathrm{t}_\mu 1\cdot \mathrm{t}_0w)=\pi(\mathrm{t}_\mu)\underbrace{\pi(\overset{\substack{W_\mathrm{a} \\ \rotatebox[origin=c]{90}{$\in$}}}{\mathrm{t}_0w})}_{=1}=\pi(\mathrm{t}_\mu)\]
and this is in $H$ because $\mu\in\Lambda(H)=\pi^{-1}(H)\cap P^\vee$, so $x\in H$. Now, if $y\in H$, then we have $y=\pi(\mathrm{t}_\mu w)$ for some $\mu\in P^\vee$ and $w\in W$ because $\pi$ is onto, but the same calculation as above shows that $y=\pi(\mathrm{t}_\mu w)=\pi(\mathrm{t}_\mu)$, so $\mu\in P^\vee\cap\pi^{-1}(H)=\Lambda(H)$ and thus $y\in\pi(W_{\Lambda(H)})=\Omega_{\Lambda(H)}$, as required.

Next, the fact that $[P^\vee:\Lambda]\cdot [\Lambda:Q^\vee]=[P^\vee:Q^\vee]=|\Omega|$ ensures that the two equalities are indeed equivalent. But since $\ker(\pi)=W_\mathrm{a}\le W_\Lambda$ we have
\[\Omega_\Lambda=\pi(W_\Lambda)=\mathrm{im}\,(W_\Lambda \stackrel{\tiny{\pi}}\to\Omega)\simeq W_\Lambda/\ker(\pi_{|W_\Lambda})=W_\Lambda/\ker(\pi)=W_\Lambda/W_\mathrm{a}\simeq \Lambda/Q^\vee.\]

Finally, using the above equality we see that the short exact sequence
\[\xymatrix{1 \ar[r] & W_\mathrm{a} \ar[r] & W_\Lambda \ar[r] & \Omega_\Lambda \ar[r] & 1}\]
is exact. It splits since the short exact sequence
\[\xymatrix{1 \ar[r] & W_\mathrm{a} \ar[r] & \widehat{W_\mathrm{a}} \ar^\pi[r] & \Omega \ar[r] & 1}\]
is split.
\end{proof}
\indent We have a useful characterization of elements of $\Omega_\Lambda$, once $\Lambda$ is known.
\begin{lem}\label{characteltsofOmega_Lambda}
Let $\Lambda$ be a $W$-lattice between $Q^\vee$ and $P^\vee$ and consider $\omega_i,\omega_j\in\Omega$. Then, the following holds
\begin{enumerate}[label=\alph*)]
\item We have $\omega_i\in\Omega_\Lambda$ if and only if $\varpi_i^\vee\in\Lambda$.
\item The group $\Omega_\Lambda$ acts on the set $\overline{\mathcal{A}_0}\cap\Lambda$.
\item We have $\omega_i\Omega_\Lambda=\omega_j\Omega_\Lambda$ if and only if $\varpi_i^\vee-\varpi_j^\vee\in\Lambda$.
\end{enumerate}
\end{lem}
\begin{proof}
\begin{enumerate}[label=\alph*)]
\item This is obvious, we have 
\[\omega_i\in\Omega_\Lambda~\Leftrightarrow~\omega_i\in W_\Lambda~\Leftrightarrow~\varpi_i^\vee=\omega_i(0)\in\Lambda.\]
\item If $\lambda\in\overline{\mathcal{A}_0}\cap\Lambda$, then in particular we have $\lambda\in\overline{\mathcal{A}_0}\cap P^\vee$ so by \cite[VI, \S 2.2, Proposition 5]{bourbaki456} there is some $k\in J$ such that $\lambda=\varpi_k^\vee\in\Lambda$. Thus, if $\omega_i\in\Omega_\Lambda$, we have $\omega_i(\lambda)=\omega_i(\varpi_k^\vee)=w_i(\varpi_k^\vee)+\varpi_i^\vee\in\Lambda$ because $\varpi_i^\vee\in\Lambda$ by $a)$ and $w_i(\varpi_k^\vee)\in\Lambda$ since $\Lambda$ is a $W$-lattice.
\item First, we compute
\[\omega_j^{-1}\omega_i=(\mathrm{t}_{\varpi_j^\vee}w_j)^{-1}\mathrm{t}_{\varpi_i^\vee}w_i=w_j^{-1}\mathrm{t}_{\varpi_i^\vee-\varpi_j^\vee}w_i=\mathrm{t}_{w_j^{-1}(\varpi_i^\vee-\varpi_j^\vee)}w_j^{-1}w_i.\]
Thus, we have
\[\omega_j^{-1}\omega_i\in\Omega_\Lambda~\Leftrightarrow~\mathrm{t}_{w_j^{-1}(\varpi_i^\vee-\varpi_j^\vee)}w_j^{-1}w_i\in\Omega_\Lambda~\Leftrightarrow~w_j^{-1}(\varpi_i^\vee-\varpi_j^\vee)\in\Lambda~\Leftrightarrow~\varpi_i^\vee-\varpi_j^\vee\in\Lambda.\]
\end{enumerate}
\end{proof}
\fi










\subsection{A $\widehat{W_\mathrm{a}}$-triangulation of $V^*$ from the barycentric subdivision an alcove}
\hfill

In order to obtain a $W_Y$-triangulation of the torus $V^*/Y$, we just have to exhibit an $\Omega_Y$-triangulation of the alcove $\overline{\mathcal{A}_0}$. As the group $\Omega_Y$ acts by affine automorphisms of $\overline{\mathcal{A}_0}$, the construction follows from the next easy result about simplicial subdivisions. 

Recall that, given a polytope $\mathcal{P}$, its \emph{barycentric subdivision} is the simplicial complex $\mathrm{Sd}(\mathcal{P})$ whose $k$-simplices are increasing chains of non-empty faces of $\mathcal{P}$ of length $k+1$. A $k$-simplex $(f_0,f_1,\dotsc,f_k)$ of $\mathrm{Sd}(\mathcal{P})$ may be geometrically realized as $\mathrm{conv}(\mathrm{bar}(f_0),\dotsc,\mathrm{bar}(f_k))$, where $\mathrm{bar}(f_i)$ stands for the barycenter of the face $f$.
\begin{lem}\label{barysubdivisequiv}
If $\mathcal{P}$ is a polytope, then $\mathrm{Sd}(\mathcal{P})$ is an $\mathrm{Aut}(\mathcal{P})$-triangulation of $\mathcal{P}$.
\end{lem}
\begin{proof}
It is well-known that $\mathrm{Sd}(\mathcal{P})$ triangulates $\mathcal{P}$ and it is clear that $\Gamma:=\mathrm{Aut}(\mathcal{P})$ permutes the simplices of $\mathrm{Sd}(\mathcal{P})$. We have to prove that, for a simplex $\sigma=(f_0,\dotsc,f_k)$ of $\mathrm{Sd}(\mathcal{P})$ and $\gamma\in\Gamma$, if $\gamma\sigma=\sigma$, then $\gamma x=x$ for each $x\in|\sigma|$.

Take $0\le i \le k$. The point $\mathrm{bar}(f_i)$ is taken by $\gamma$ to some $\mathrm{bar}(f_j)$ and since the barycenter of a polytope lies in its relative interior, we have $\gamma(\mathring{f_i})\cap\mathring{f_j}\ne\emptyset$ (where $\mathring{\cdot}$ is the relative interior) and as $\gamma$ acts as an automorphism of $\mathcal{P}$, this forces $\gamma(f_i)=f_j$ and $\dim(f_i)=\dim(\gamma(f_i))=\dim(f_j)$. But the sequence $(\dim f_0,\dotsc,\dim f_k)$ is increasing, so $f_i=f_j$ and $\mathrm{bar}(f_i)=\mathrm{bar}(f_j)=\gamma(\mathrm{bar}(f_i))$. The conclusion now follows from the equality $|\sigma|=\mathrm{conv}(\mathrm{bar}(f_0),\dotsc,\mathrm{bar}(f_k))$.
\end{proof}


\begin{center}\label{extendeddynkindiagrams}
\resizebox{0.8\textwidth}{!}{
\begin{tabular}{|c|c|c|}
\hline
Type & Extended Dynkin diagram & Fundamental group $\Omega\le\mathrm{Aut}({\mathcal{D}ynkin_0})$ \\
\hline
\hline
$\widetilde{A_1}$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$0$};
	
	\draw (a)--(b);
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=white] (b) circle (2.5pt);
	\node[mark size=2.5pt] at (b) {\pgfuseplotmark{otimes}};
	\draw (a) circle (2.5pt);
	
	\draw (1/2,0) node[above]{$\infty$};
\end{tikzpicture}\end{minipage} & $\omega_1=(0,1)$ \\
\hline
 $\widetilde{A_n}~(n\ge2)$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-2,0);
	\coordinate (b) at (-1,0);
	\coordinate (c) at (0,0);
	\coordinate (d) at (1,0);
	\coordinate (e) at (2,0);
	\coordinate (f) at (0,1);
	\coordinate (bc) at (-1/2,0);
	\coordinate (cd) at (1/2,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node{$\cdots$};
	\draw (d) node[below]{${n-1}$};
	\draw (e) node[below]{$n$};
	\draw (f) node[above]{$0$};
	
	\draw (a)--(b)
	(b)--(bc)
	(cd)--(d)
	(d)--(e)
	(e)--(f)
	(f)--(a);
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=white] (b) circle (2.5pt);
	\fill[fill=white] (d) circle (2.5pt);
	\fill[fill=white] (e) circle (2.5pt);
	\fill[fill=white] (f) circle (2.5pt);
	
	\draw (a) circle (2.5pt);
	\draw (b) circle (2.5pt);
	\draw (d) circle (2.5pt);
	\draw (e) circle (2.5pt);
	\node[mark size=2.5pt] at (f) {\pgfuseplotmark{otimes}};
\end{tikzpicture}\end{minipage} & $\begin{array}{ll} \omega_{1}=(0,1,2,\cdots,n) \\ \\ \omega_{i}=(\omega_{1})^i,~0\le i \le n\end{array}$ \\
\hline
 & & \\
$\widetilde{B_2}=\widetilde{C_2}$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (0,0);
	\coordinate (e) at (1,0);
	\coordinate (f) at (2,0);
	
	\draw (a) node[below]{$0$};
	\draw (e) node[below]{$1$};
	\draw (f) node[below]{$2$};
	
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (a)--(e);
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (f)--(e);
	
	\fill[fill=white] (a) circle (2.5pt);
	\node[mark size=2.5pt] at (a) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=white] (f) circle (2.5pt);
    \draw (f) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\omega_{1}=(0,2)$ \\
 & & \\
\hline
$\widetilde{B_n}~(n\ge3)$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-1/2,1.732/2);
	\coordinate (b) at (-1/2,-1.732/2);
	\coordinate (c) at (1/2,0);
	\coordinate (d) at (1+1/2,0);
	\coordinate (e) at (2+1/2,0);
	\coordinate (f) at (3+1/2,0);
	\coordinate (g) at (4+1/2,0);
	
	\draw (a) node[left]{$1$};
	\draw (b) node[left]{$0$};
	\draw (c) node[below]{$2$};
	\draw (d) node[below]{$3$};
	\draw (e) node{$\cdots$};
	\draw (f) node[below]{${n-1}$};
	\draw (g) node[below]{$n$};
	
	\draw (a)--(c)
	(b)--(c)
	(c)--(d)
	(d)--(2,0)
	(3,0)--(f);
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (f)--(g);
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=black] (g) circle (2.5pt);
	\fill[fill=white] (b) circle (2.5pt);
	\node[mark size=2.5pt] at (b) {\pgfuseplotmark{otimes}};
	\draw (a) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\omega_{1}=(0,1)$ \\
\hline
 & & \\
$\widetilde{C_n}~(n\ge3)$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-3,0);
	\coordinate (b) at (-2,0);
	\coordinate (c) at (-1,0);
	\coordinate (d) at (0,0);
	\coordinate (e) at (1,0);
	\coordinate (f) at (2,0);
	
	\draw (a) node[below]{$0$};
	\draw (b) node[below]{$1$};
	\draw (c) node[below]{$2$};
	\draw (d) node{$\cdots$};
	\draw (e) node[below]{${n-1}$};
	\draw (f) node[below]{$n$};
	
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (a)--(b);
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (f)--(e);
	\draw (b)--(c)
	(c)--(-1/2,0)
	(1/2,0)--(e);
	
	\fill[fill=white] (a) circle (2.5pt);
	\node[mark size=2.5pt] at (a) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=white] (f) circle (2.5pt);
	
    \draw (f) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\displaystyle{\omega_{n}=(0,n)\prod_{i=1}^{\left\lfloor \frac{n-1}{2}\right\rfloor}(i,{n-i})}$ \\
 & & \\
\hline
 & & \\
$\widetilde{D_{2n}}~(n\ge2)$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-1/2,1.732/2);
	\coordinate (b) at (-1/2,-1.732/2);
	\coordinate (c) at (1/2,0);
	\coordinate (d) at (1+1/2,0);
	\coordinate (e) at (2+1/2,0);
	\coordinate (f) at (3+1/2,0);
	\coordinate (g) at (4+1/2,1.732/2);
	\coordinate (h) at (4+1/2,-1.732/2);
	
	\draw (a) node[left]{$1$};
	\draw (b) node[left]{$0$};
	\draw (c) node[below]{$2$};
	\draw (d) node[below]{$3$};
	\draw (e) node{$\cdots$};
	\draw (f) node[right]{${2n-2}$};
	\draw (g) node[right]{${2n}$};
	\draw (h) node[right]{${2n-1}$};
	
	\draw (a)--(c)
	(b)--(c)
	(c)--(d)
	(d)--(2,0)
	(3,0)--(f)
	(f)--(g)
	(f)--(h);
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=white] (g) circle (2.5pt);
	\fill[fill=white] (b) circle (2.5pt);
	\fill[fill=white] (h) circle (2.5pt);
	\node[mark size=2.5pt] at (b) {\pgfuseplotmark{otimes}};
	\draw (a) circle (2.5pt);
	\draw (g) circle (2.5pt);
	\draw (h) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\displaystyle{\begin{array}{llll} \omega_{1}=(0,1)({2n-1},{2n}) \\ \\ \omega_{{2n-1}}=(0,{2n-1})(1,{2n})\prod_{i=2}^{n-1}(i,{2n-i}) \\ \\ \omega_{{2n}}=(0,{2n})(1,{2n-1})\prod_{i=2}^{n-1}(i,{2n-i})=\omega_{1}\omega_{{2n-1}}\end{array}}$ \\
 & & \\
\hline
 & & \\
$\widetilde{D_{2n+1}}~(n\ge2)$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-1/2,1.732/2);
	\coordinate (b) at (-1/2,-1.732/2);
	\coordinate (c) at (1/2,0);
	\coordinate (d) at (1+1/2,0);
	\coordinate (e) at (2+1/2,0);
	\coordinate (f) at (3+1/2,0);
	\coordinate (g) at (4+1/2,1.732/2);
	\coordinate (h) at (4+1/2,-1.732/2);
	
	\draw (a) node[left]{$1$};
	\draw (b) node[left]{$0$};
	\draw (c) node[below]{$2$};
	\draw (d) node[below]{$3$};
	\draw (e) node{$\cdots$};
	\draw (f) node[right]{${2n-1}$};
	\draw (g) node[right]{${2n+1}$};
	\draw (h) node[right]{${2n}$};
	
	\draw (a)--(c)
	(b)--(c)
	(c)--(d)
	(d)--(2,0)
	(3,0)--(f)
	(f)--(g)
	(f)--(h);
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=white] (g) circle (2.5pt);
	\fill[fill=white] (b) circle (2.5pt);
	\fill[fill=white] (h) circle (2.5pt);
	\node[mark size=2.5pt] at (b) {\pgfuseplotmark{otimes}};
	\draw (a) circle (2.5pt);
	\draw (g) circle (2.5pt);
	\draw (h) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\displaystyle{\begin{array}{llll} \omega_{1}=(0,1)({2n},{2n+1}) \\ \\ \omega_{{2n}}=(0,{2n},1,{2n+1})\prod_{i=2}^{n}(i,{2n+1-i}) \\ \\ \omega_{{2n+1}}=(0,{2n+1},1,{2n})\prod_{i=2}^{n}(i,{2n+1-i})\end{array}}$ \\
 & & \\
\hline
$\widetilde{E_6}$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-2,0);
	\coordinate (b) at (-1,0);
	\coordinate (c) at (0,0);
	\coordinate (d) at (1,0);
	\coordinate (e) at (2,0);
	\coordinate (f) at (0,1);
	\coordinate (g) at (0,2);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$4$};
	\draw (d) node[below]{$5$};
	\draw (e) node[below]{$6$};
	\draw (f) node[right]{$2$};
	\draw (g) node[right]{$0$};
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d)
	(d)--(e)
	(c)--(f)
	(f)--(g);
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=white] (e) circle (2.5pt);
	\fill[fill=white] (g) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\draw (a) circle (2.5pt);
	\draw (e) circle (2.5pt);
	\node[mark size=2.5pt] at (g) {\pgfuseplotmark{otimes}};
\end{tikzpicture}\end{minipage} & $\begin{array}{ll} \omega_{1}=(0,1,6)(2,3,5) \\ \\ \omega_{6}=(1,0,6)(3,2,5)=\omega_{1}^{-1}\end{array}$ \\
\hline
$\widetilde{E_7}$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-2,0);
	\coordinate (b) at (-1,0);
	\coordinate (c) at (0,0);
	\coordinate (d) at (1,0);
	\coordinate (e) at (2,0);
	\coordinate (f) at (0,1);
	\coordinate (g) at (3,0);
	\coordinate (h) at (-3,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$4$};
	\draw (d) node[below]{$5$};
	\draw (e) node[below]{$6$};
	\draw (f) node[right]{$2$};
	\draw (g) node[below]{$7$};
	\draw (h) node[below]{$0$};
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d)
	(d)--(e)
	(c)--(f)
	(e)--(g)
	(h)--(a);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=white] (g) circle (2.5pt);
	\fill[fill=white] (h) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\draw (g) circle (2.5pt);
	\node[mark size=2.5pt] at (h) {\pgfuseplotmark{otimes}};
\end{tikzpicture}\end{minipage} & $\omega_{7}=(0,7)(1,6)(3,5)$ \\
\hline
$\widetilde{E_8}$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-2,0);
	\coordinate (b) at (-1,0);
	\coordinate (c) at (0,0);
	\coordinate (d) at (1,0);
	\coordinate (e) at (2,0);
	\coordinate (f) at (0,1);
	\coordinate (g) at (3,0);
	\coordinate (h) at (5,0);
	\coordinate (i) at (4,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$4$};
	\draw (d) node[below]{$5$};
	\draw (e) node[below]{$6$};
	\draw (f) node[right]{$2$};
	\draw (g) node[below]{$7$};
	\draw (h) node[below]{$0$};
	\draw (i) node[below]{$8$};
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d)
	(d)--(e)
	(c)--(f)
	(e)--(g)
	(h)--(i)
	(g)--(i);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=black] (g) circle (2.5pt);
	\fill[fill=white] (h) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=black] (i) circle (2.5pt);
	\node[mark size=2.5pt] at (h) {\pgfuseplotmark{otimes}};
\end{tikzpicture}\end{minipage} & $\varnothing$ \\
\hline
 & & \\
\centering{$\widetilde{F_4}$} & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-2,0);
	\coordinate (b) at (-1,0);
	\coordinate (c) at (0,0);
	\coordinate (d) at (1,0);
	\coordinate (e) at (2,0);
	
	\draw (a) node[below]{$0$};
	\draw (b) node[below]{$1$};
	\draw (c) node[below]{$2$};
	\draw (d) node[below]{$3$};
	\draw (e) node[below]{$4$};
	
	\draw (a)--(b)
	(b)--(c)
	(d)--(e);	
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (c)--(d);
	
	\fill[fill=white] (a) circle (2.5pt);
	\node[mark size=2.5pt] at (a) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\varnothing$ \\
\hline
 & & \\
$\widetilde{G_2}$ & \begin{minipage}{0.5\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$0$};
	
	\draw (b)--(c);
	\draw[decoration={markings, mark=at position 0.75 with {\arrow{angle 60}}},postaction={decorate},double distance=2.5pt] (b)--(a);
	\draw (b)--(a);
	
	\fill[fill=white] (c) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\varnothing$ \\
\hline
\end{tabular}}
\captionof{table}{Extended Dynkin diagrams and fundamental groups elements. \\ The white dots stand for the roots corresponding to minuscule weights and the crossed dots represent the lowest root $\widetilde{\alpha}:=-\alpha_0$.}
\end{center}



From this we deduce that $W_\mathrm{a}\cdot\mathrm{Sd}(\overline{\mathcal{A}_0})$ is a $W_Y$-triangulation of $V^*$ for all $Q^\vee\subset Y\subset P^\vee$ at once. We can describe the associated ${\mathbb Z}[W_Y]$-complex $C^*_\mathrm{cell}(V^*,W_Y;{\mathbb Z})$ using the face lattice of $\mathrm{Sd}(\overline{\mathcal{A}_0})$, but the description is rather tedious and not as nice as for the simply-connected case, as the combinatorics of parabolic subgroups doesn't make sense anymore.


There is a bijection $\mathrm{vert}(\overline{\mathcal{A}_0})\approx S_0=\{0,\dotsc,n\}$ and $\overline{\mathcal{A}_0}\simeq\Delta^n$, so that the face lattice of $\overline{\mathcal{A}_0}$ is $F(\overline{\mathcal{A}_0})=(\mathscr{P}(S_0),\subset)$. This gives a description of the face lattice of $\mathrm{Sd}(\overline{\mathcal{A}_0})$ as
\[\forall 0 \le d \le n,~F_d(\mathrm{Sd}(\overline{\mathcal{A}_0}))=\{Z_\bullet=(Z_0,Z_1,\dotsc,Z_d)~;~\forall i,~\emptyset\ne Z_i\subset S_0,~Z_i\subsetneq Z_{i+1}\}\]
and $Z_\bullet\subset Z_\bullet'$ if $Z_\bullet$ is a subsequence of $Z_\bullet'$.
\begin{lem}\label{decofstabs}
The group $\Omega_Y$ acts on $\overline{\mathcal{A}_0}$ and this induces an action on $S_0$. The resulting action on $F(\mathrm{Sd}(\overline{\mathcal{A}_0}))$ corresponds to the action of $\Omega_Y$ on $|\mathrm{Sd}(\overline{\mathcal{A}_0})|=\overline{\mathcal{A}_0}$.

Moreover, for $Z_\bullet\in F_d(\mathrm{Sd}(\overline{\mathcal{A}_0}))$, the stabilizer of $Z_\bullet$ in $W_Y$ decomposes as
\[(W_Y)_{Z_\bullet}=(W_\mathrm{a})_{Z_\bullet}\rtimes (\Omega_Y)_{Z_\bullet}=(W_\mathrm{a})_{S_0\setminus Z_d}\rtimes (\Omega_Y)_{Z_\bullet},\]
and 
\[(\Omega_Y)_{Z_\bullet}=\bigcap_{i=0}^d\Omega_{Z_i}.\]
\end{lem}
\begin{proof}
The first statement is obvious. Write $Z_\bullet=(Z_0\subsetneq\cdots\subsetneq Z_d)$ and let $\widehat{w}:=w\omega_j\in(W_Y)_{Z_\bullet}$ with $w\in W_\mathrm{a}$ and $\omega_j\in\Omega_Y$. Then, for every $x\in|Z_{\bullet}|$, we have $\widehat{w}(x)=w(\omega_j(x))=x$ and $\omega_j(x)\in\overline{\mathcal{A}_0}$ so $x=\omega_j(x)$ and $\omega_j\in(\Omega_Y)_{Z_\bullet}$. On another hand we get $w(x)=x$ so $w\in(W_\mathrm{a})_{Z_\bullet}$.

Now, an element $w\in W_\mathrm{a}$ fixes $Z_\bullet$ if and only if it fixes the maximal face of $Z_\bullet$, i.e. $Z_d$. This is indeed the parabolic subgroup $(W_\mathrm{a})_{S_0\setminus Z_d}$. The last equality holds in general and is straightforward.
\end{proof}

To avoid too many choices, we fix a total ordering $\prec$ on $F(\mathrm{Sd}(\overline{\mathcal{A}_0}))$. For instance, the lexicographical order $<_{lex}$ induced by the order on $\mathscr{P}(S_0)=2^{S_0}$ inherited from the natural order on $S_0$.

As the barycentric subdivision of $\overline{\mathcal{A}_0}$ is simplicial, the boundaries of the complex and the cup product are easily determined and lead to the following result:
\begin{theo}\label{complexforY}
For $0\le d\le n$, decompose the $\Omega_Y$-set $F_d(\mathrm{Sd}(\overline{\mathcal{A}_0}))$ into orbits
\[F_d(\mathrm{Sd}(\overline{\mathcal{A}_0}))/\Omega_Y\approx\{Z_{d,1}\prec\cdots\prec Z_{d,k_d}\},~~\text{where}~~Z_{d,i}=\min_{\prec}(\Omega_Y\cdot Z_{d,i}).\]
Denote further
\[\forall 0\le p \le d,~\forall 1\le i \le k_d,~Z_{d,i}^{(p)}:=((Z_{d,i})_0,\dotsc,\widehat{(Z_{d,i})_p},\dotsc,(Z_{d,i})_d).\]
Then the complex $C^\mathrm{cell}_*(V^*,W_Y;{\mathbb Z})$ is given by
\[C^\mathrm{cell}_d(V^*,W_Y;{\mathbb Z})=\bigoplus_{i=1}^{k_d}{\mathbb Z}\left[W_Y/(W_Y)_{Z_{d,i}}\right],\]
with 
\[(W_Y)_{Z_{d,i}}=(W_\mathrm{a})_{(Z_{d,i})_d^\complement}\rtimes \bigcap_{j=0}^d(\Omega_Y)_{(Z_{d,i})_j}.\]
The boundaries are given by
\[\partial_d(Z_{d,i})=\sum_{p=0}^d(-1)^p\omega_{p,i}(Z_{d-1,u_i}),~\text{where }u_i\in S_0~;~Z_{d-1,u_i}=\min_\prec(\Omega_Y\cdot Z_{d,i}^{(p)})~\text{and}~\omega_{p,i}(Z_{d-1,u_i})=Z_{d,i}^{(p)}.\]

Moreover, the dual complex $C^*_\mathrm{cell}(V^*,\widehat{W_\mathrm{a}};{\mathbb Z})$ is a ${\mathbb Z}[W_Y]$-dg-algebra with product
\[Z_{d,i}^*\cup Z_{e,j}^*=\delta_{(Z_{d,i})_d,(Z_{e,j})_0}\omega(Z_{d+e,k})^*,\]
where
\[Z_{d+e,k}=\min_\prec(\Omega_Y\cdot((Z_{d,i})_0,\dotsc,(Z_{d,i})_d,(Z_{e,j})_0,\dotsc,(Z_{e,j})_e))~~\text{and}~~\omega(Z_{d+e,k})=((Z_{d,i})_0,\dotsc,(Z_{e,j})_e).\]

Finally, the complex for the torus $V^*/Y$ is given by
\[C^\mathrm{cell}_*(V^*/Y,W;{\mathbb Z})=\mathrm{Def}_W^{W_Y}(C_*^\mathrm{cell}(V^*,W_Y;{\mathbb Z})).\]
\end{theo}
\begin{exemple}\label{A2example}
The case of $A_2$ is fairly computable by hand. We denote by $\Phi=\{\pm\alpha,\pm\beta,\pm(\alpha+\beta)\}$ a root system of type $A_2$, with simple system $\Pi=\{\alpha,\beta\}$, so $S_0=\{0,1,2\}=J$ and 
\[\Omega=\Omega_{P^\vee}=\{1,\underbrace{t_{\varpi_\alpha^\vee}s_\alpha s_\beta}_{\omega_\alpha},\underbrace{t_{\varpi_\beta^\vee}s_\beta s_\alpha}_{\omega_\beta}\}\simeq {\mathbb Z}/3{\mathbb Z}.\]
In this case, $W_{P^\vee}=\widehat{W_\mathrm{a}}$ is the classical extended affine Weyl group.

Geometrically, the element $\Omega_\alpha$ acts as the rotation with angle $2\pi/3$ around the barycenter of $\overline{\mathcal{A}_0}=\mathrm{conv}(0,\varpi_\alpha^\vee,\varpi_\beta^\vee)=:[0,1,2]\simeq\Delta^2$. The situation can be visualized in Figure \ref{alcoveA2}.
\begin{figure}[h!]
\begin{tikzpicture}[scale=2,rotate=45]  
  \coordinate (z) at (0,0);
  \coordinate (a) at (1,-1);
  \coordinate (b) at (0.3660254040,1.366025404);
  \coordinate (ab) at (1+0.3660254040,-1+1.366025404);
  \coordinate (ma) at (-1,1);
  \coordinate (mb) at (-0.3660254040,-1.366025404);
  \coordinate (mab) at (-1-0.3660254040,1-1.366025404);
  
  \coordinate (la) at (2/3+1/3*0.3660254040,-2/3+1/3*1.366025404);
  \coordinate (lb) at (1/3+2/3*0.3660254040,-1/3+2/3*1.366025404);
  
  \coordinate (lad) at (1/3+1/6*0.3660254040,-1/3+1/6*1.3660254040);
  \coordinate (lbd) at (1/6+1/3*0.3660254040,-1/6+1/3*1.3660254040);
  \coordinate (barlab) at (1/3+1/3*0.3660254040,-1/3+1/3*1.3660254040);
  
  \fill[fill=black] (a) circle (1pt);
  \fill[fill=black] (b) circle (1pt);
  \fill[fill=black] (ab) circle (1pt);
  \fill[fill=black] (ma) circle (1pt);
  \fill[fill=black] (mb) circle (1pt);
  \fill[fill=black] (mab) circle (1pt);
  
  \fill[fill=red] (la) circle (1pt);
  \fill[fill=red] (lb) circle (1pt);
  \fill[fill=red] (z) circle (1pt);
  
  \draw (z)--(a);
  \draw (z)--(b);
  \draw (z)--(ab);
  \draw[dashed] (z)--(ma);
  \draw[dashed] (z)--(mb);
  \draw[dashed] (z)--(mab);
  
  \draw (a) node[right]{$\alpha^\vee$};
  \draw (b) node[left]{$\beta^\vee$};
  \draw (ab) node[above]{$\alpha_0^\vee=\alpha^\vee+\beta^\vee$};
  \draw (la) node[right]{$\varpi_\alpha^\vee$};
  \draw (lb) node[left]{$\varpi_\beta^\vee$};
  \draw (0,0.05) node[below left]{$0$};
  
  \fill[fill=blue,opacity=0.55] (z)--(la)--(lb);
  
  \draw (z)--(la);
  \draw (z)--(lb);
  \draw (la)--(lb);
  \draw (la)--(lbd);
  \draw (lb)--(lad);
  \draw (z)--(ab);
  
  \coordinate (x) at (2+0.3660254040,-2+1.3660254040);
  \coordinate (y) at (1+2*0.3660254040,-1+2*1.3660254040);

\end{tikzpicture}
\caption{The subdivision $|\mathrm{Sd}(\overline{\mathcal{A}_0})|$ of the fundamental alcove $\overline{\mathcal{A}_0}$ (in blue).}
\label{alcoveA2}
\end{figure}

There are three $\widehat{W_\mathrm{a}}$-orbits of points in $|\mathrm{Sd}(\overline{\mathcal{A}_0})|$ and we represent them by the points
\[e^0_1:=(\{0\})=0,~e^0_2:=(\{0,1\})=\frac{\varpi_\alpha^\vee}{2},~e^0_3:=(\{0,1,2\})=\frac{\varpi_\alpha^\vee+\varpi_\beta^\vee}{3}.\]
Remember that we order $\mathscr{P}(S_0)$ lexicographically and these are lex-minimal in their orbits. There are also four orbits of 1-cells represented by
\[e^1_1:=(\{0\},\{0,1\}),~e^1_2:=(\{0\},\{0,2\}),~e^1_3:=(\{0\},\{0,1,2\}),~e^1_4:=(\{0,1\},\{0,1,2\}).\]
Finally, there are two orbits of 2-cells represented by
\[e^2_1:=(\{0\},\{0,1\},\{0,1,2\}),~e^2_2:=(\{0\},\{0,2\},\{0,1,2\}).\]
Now, we have
\[\forall e\in\{e^0_1,e^0_2,e^1_i,e^2_j\},~\Omega_e=1~~\text{and}~~\Omega_{e^0_3}=\Omega\]
and we obtain the non-trivial stabilizers in $\widehat{W_\mathrm{a}}$:
\[(\widehat{W_\mathrm{a}})_{e^0_3}=\Omega,~(\widehat{W_\mathrm{a}})_{e^0_1}=W,~(\widehat{W_\mathrm{a}})_{e^0_2}=(\widehat{W_\mathrm{a}})_{e^1_1}=\left<s_\beta\right>,(\widehat{W_\mathrm{a}})_{e^1_2}=\left<s_\alpha\right>.\]
The boundaries are readily computed, with for instance
\[\partial_2(e^2_2)=e^1_2-e^1_3+(\{0,2\},\{0,1,2\})=e^1_2-e^1_3+\omega_\beta e^1_4.\]

Therefore, the complex $C^\mathrm{cell}_*(V^*,\widehat{W_\mathrm{a}};{\mathbb Z})$ is given by
\[\xymatrix{{\mathbb Z}[\widehat{W_\mathrm{a}}]^2 \ar^<<<<<{\partial_2}[r] & {\mathbb Z}[\widehat{W_\mathrm{a}}/\left<s_\beta\right>]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}/\left<s_\alpha\right>]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}]^2 \ar^{\partial_1}[r] & {\mathbb Z}[\widehat{W_\mathrm{a}}/W]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}/\left<s_\beta\right>]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}/\Omega]},\]
with
\[\partial_2=\begin{pmatrix}1 & 0 & -1 & 1 \\ 0 & 1 & -1 & \omega_\beta\end{pmatrix},~~\partial_1=\begin{pmatrix}-1 & 1 & 0 \\ -1 & \omega_\beta & 0 \\ -1 & 0 & 1 \\ 0 & -1 & 1\end{pmatrix}.\]
Moreover, the root datum $(P,\Phi,P^\vee,\Phi^\vee)$ may be realized by the Lie group $PSU(3)=SU(3)/\mu_3$ with torus $T=T_0/\mu_3\simeq V^*/P^\vee$, where $T_0$ is the standard torus consisting of diagonal matrices of $SU(3)$. The complex 
\[C^\mathrm{cell}_*(T,W;{\mathbb Z})=\mathrm{Def}^{\widehat{W_\mathrm{a}}}_W(C^\mathrm{cell}_*(V^*,\widehat{W_\mathrm{a}};{\mathbb Z}))\]
then becomes
\[\xymatrix{{\mathbb Z}[W]^2 \ar^<<<<<{\overline{\partial_2}}[r] & {\mathbb Z}[W/\left<s_\beta\right>]\oplus{\mathbb Z}[W/\left<s_\alpha\right>]\oplus{\mathbb Z}[W]^2 \ar^<<<<<{\overline{\partial_1}}[r] & {\mathbb Z}\oplus{\mathbb Z}[W/\left<s_\beta\right>]\oplus{\mathbb Z}[W/\left<s_\alpha s_\beta\right>]},\] 
with
\[\overline{\partial_2}=\begin{pmatrix}1 & 0 & -1 & 1 \\ 0 & 1 & -1 & s_\beta s_\alpha \end{pmatrix},~~\overline{\partial_1}=\begin{pmatrix}-1 & 1 & 0 \\ -1 & s_\beta s_\alpha & 0 \\ -1 & 0 & 1 \\ 0 & -1 & 1\end{pmatrix}.\]

Recall that the complex $C^\mathrm{cell}_*(T_0,W;{\mathbb Z})$ is the simpler one
\[\xymatrix{{\mathbb Z}[W] \ar^>>>>>>>>>>{\tiny{\begin{pmatrix}1 & 1 & -1\end{pmatrix}}}[rr] & & {\mathbb Z}[W/\left<s_\beta\right>]\oplus{\mathbb Z}[W/\left<s_\alpha s_\beta s_\alpha\right>]\oplus{\mathbb Z}[W/\left<s_\alpha\right>]\ar^<<<<<<<<<{\tiny{\begin{pmatrix}-1 & 1 & 0 \\ 0 & -1 & 1 \\ -1 & 0 & 1\end{pmatrix}}}[rr] & & {\mathbb Z}^3}.\]

The complexes $C^\mathrm{cell}_*(T_0,W;{\mathbb Z})$ and $C^\mathrm{cell}_*(T,W;{\mathbb Z})$ may be obtained using the commands 
\newline
\emph{\texttt{ComplexForFiniteCoxeterGroup("A",2)}}and \emph{\texttt{CellularComplexT("A",2,[0,1,2])}} provided by the package \emph{\texttt{Salvetti-and-tori-complexes}}, mentioned in the Introduction.
\end{exemple}
\begin{rem}
The complex $C^\mathrm{cell}_*(V^*,\widehat{W_\mathrm{a}};{\mathbb Z})$ in the previous example can be reduced. Indeed, we can take $e^2:=e^2_1\cup_{e^1_3}e^2_2$ as 1-cell. This deletes the 1-cell $e^1_3$ and the complex reduces to

\[\xymatrix{{\mathbb Z}[\widehat{W_\mathrm{a}}] \ar^<<<<{\tiny{{}^t{\left(\begin{smallmatrix}1 \\ -1 \\ 1-\omega_\beta\end{smallmatrix}\right)}}}[r] & {\mathbb Z}[\widehat{W_\mathrm{a}}/\left<s_\beta\right>]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}/\left<s_\alpha\right>]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}] \ar^{\tiny{\left(\begin{smallmatrix}-1 & 1 & 0 \\ -1 & \omega_\beta & 0 \\ 0 & -1 & 1\end{smallmatrix}\right)}}[rr] & & {\mathbb Z}[\widehat{W_\mathrm{a}}/W]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}/\left<s_\beta\right>]\oplus{\mathbb Z}[\widehat{W_\mathrm{a}}/\Omega]}.\]


We recognize the closure $\overline{e^2}=\mathrm{conv}(e^0_1,e^0_2,\omega_\beta e^0_2,e^0_3)$ as the fundamental polytope for $\widehat{W_\mathrm{a}}$ discovered by Komrakov and Premet in \cite{komrakov-premet}. It would be nice to obtain a triangulation from this polytope. However, this approach violently fails in general. For instance, even in type $C_3$ (and in fact in every non simply-connected root system of rank at least 3, except $B_n$), there is a face of this polytope whose vertices are non-trivially permuted by the action of $\Omega$ and thus the resulting triangulation is not $\Omega$-equivariant.
\end{rem}









































\part{Hyperbolic tori for non-crystallographic Coxeter groups}
\section{Construction of the hyperbolic extensions and the hyperbolic torus}


The goal of this part is to construct a smooth manifold affording the above dg-algebra and playing the role of a torus for non-crystallographic groups. All such groups are listed in the following table:
\medskip
\begin{center}\label{noncrystcoxgroups}
\resizebox{0.55\textwidth}{!}{
\begin{tabular}{|c|c|}
\hline
Type & Coxeter diagram \\
\hline
\hline
$I_2(m)~(5\le m\ne 6)$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (-1/2,0) node[above]{$m$};
	
	\draw (a)--(b);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
$H_3$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$3$};
	\draw (-1/2,0) node[above]{$5$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
$H_4$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	\coordinate (d) at (2,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$3$};
	\draw (d) node[below]{$4$};
	\draw (-1/2,0) node[above]{$5$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	\draw (c)--(d);
	
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
\end{tabular}}
\captionof{table}{Coxeter diagrams of finite non-crystallographic Coxeter systems.}
\end{center}
\bigskip

Although we shall focus on the non-crystallographic case, what follows applies to all finite irreducible Coxeter groups. In particular, in the $I_2(m)$ case, we only assume that $m\ge 3$.

A key ingredient in the construction of the $W$-equivariant triangulation of the torus of a Weyl group (in fact of the simply-connected compact Lie group of type $W$) is the reflection $s_0$ associated to the highest root $\alpha_0$ in the root system $\Phi$ of $W$. Thus, we first have to find such a reflection in the non-crystallographic cases. Moreover, this should come with an infinite extension of the finite group, which should again be a Coxeter group. A last important point is that, in the infinite Coxeter group, every maximal proper parabolic subgroup should be finite. The good class of Coxeter groups we shall consider for this matter is no longer the affine groups, but the compact hyperbolic groups.








\subsection{Compact hyperbolic extensions of $I_2(m)$, $H_3$ and $H_4$}
\hfill

Let us first recall some basic terminology concerning Coxeter groups. For more detailed discussions, the reader is referred to \cite{bourbaki456} and \cite{humphreys-reflectiongroups}.

Let $(W,S)$ be an irreducible Coxeter system of rank $n$. We write
\[W=\left<s_1,\dotsc,s_n~|~(s_is_j)^{m_{i,j}}=1\right>,\]
with $M=(m_{i,j})_{1\le i,j \le n}$ the Coxeter matrix of $(W,S)$. Recall (\cite[V, \S 4]{bourbaki456} or \cite[Chap. 5]{humphreys-reflectiongroups}) that on the formal vector space $V:=\mathrm{span}_{{\mathbb R}}(\alpha_i,~1\le i \le n)$ we may define a symmetric bilinear form by
\[B(\alpha_i,\alpha_j):=-\cos\left(\frac{\pi}{m_{i,j}}\right)\]
as well as the linear mappings
\[\forall 1 \le i \le n,~\sigma_i:=(v\mapsto v-2B(\alpha_i,v)\alpha_i).\]
Then the assignment $s_i \mapsto \sigma_i$ extends uniquely to a faithful irreducible representation
\[\sigma : W \longrightarrow GL(V),\]
known as the \emph{geometric representation} of $W$. 

Moreover, $W$ is finite (resp. affine) if and only if the form $B$ is positive definite (resp. positive semidefinite) (see \cite[V, \S 4.8 and 4.9]{bourbaki456}).

\begin{prop-def}[{\cite[\S 6.8]{humphreys-reflectiongroups}}]\label{hyperbolics}
The followings are equivalent
\begin{enumerate}[label=(\roman*)]
\item The form $B$ has signature $(n-1,1)$ and $B(\lambda,\lambda)<0$ for $\lambda\in C$,
\item The form $B$ is non-degenerate but not positive and the graph obtained by removing any vertex from the graph of $W$ is of non-negative type (i.e. its group is finite or affine).
\end{enumerate}

If these conditions occur, then $W$ is said to be \emph{hyperbolic}. If the second condition is enhanced by requiring that any such sub-graph is of positive definite type (i.e. its group is finite), then $W$ is said to be \emph{compact hyperbolic}.
\end{prop-def}
\begin{rem}\label{hyperfromhum}
As mentioned in \cite{humphreys-reflectiongroups}, the terminology comes from the fact that the homogeneous space $O(V,B)/W$, equipped with the induced measure coming from the Haar measure on $O(V,B)$, is of finite volume if and only if $W$ is finite or hyperbolic and, in the hyperbolic case, a component of $\{\lambda\in V~;~B(\lambda,\lambda)=-1\}$ gives a model for the hyperbolic $(n-1)$-space $\mathbb{H}^n$. Moreover, the space $O(V,B)/W$ is compact if and only if $W$ is compact hyperbolic.

Another point is that $W$ is compact hyperbolic if and only if $B$ is non-degenerate non-positive and every proper parabolic subgroup of $W$ is finite.
\end{rem}
We are ready to define the compact hyperbolic extensions of non-crystallographic groups. From now on, we let $(W,S)$ be a finite irreducible Coxeter system of rank $n$ with Coxeter matrix $M$. As mentioned in the beginning of this part, if $W$ is crystallographic, then we consider the reflection $r_W:=s_{\alpha_0}$ associated to the highest root $\alpha_0$ of $\Phi(W)$ and, if $W$ is non-crystallographic, we have to choose a suitable reflection $r_W\in W$ yielding a compact hyperbolic extension of $W$. This is done in the following result, where the notation are as in the Table \ref{noncrystcoxgroups}.
\begin{prop-def}\label{cpcthypextsofnoncryst}
Let $W$ be non-crystallographic and choose $r_W\in W$ to be the following reflection in $W$:
\[r_W:=\left\{\begin{array}{ccc}
(s_1s_2)^{\left\lfloor\frac{m-1}{2}\right\rfloor}s_1 & \text{if} & W=I_2(m),~m\ge 3, \\[.5em]
s_3^{(s_2s_1)^2} & \text{if} & W=H_3, \\[.5em]
s_4^{\left(s_1^{s_2s_3}(s_1s_2)^2s_3s_4\right)^2} & \text{if} & W=H_4. \end{array}\right.\]

Define
\[\widehat{W}:=\left<\widehat{s}_0,\widehat{s}_1,\dotsc,\widehat{s}_r~\left|~\forall i,j\ge1,~(\widehat{s}_i\widehat{s}_j)^{m_{i,j}}=(\widehat{s}_0\widehat{s}_i)^{o(r_Ws_i)}={\widehat{s}_0}^2=1\right.\right>,\]
where $o(x)$ is the order of the element $x$ and $\widehat{S}:=\{\widehat{s}_0,\dotsc,\widehat{s}_n\}$. Then the pair $(\widehat{W},\widehat{S})$ is a compact hyperbolic Coxeter system, whose Coxeter graph is as in the following table:
\medskip
\begin{table}[h!]
\resizebox{0.45\textwidth}{!}{
\begin{tabular}{|c|c|}
\hline
Extension & Coxeter graph \\
\hline
\hline
$\widehat{I_2(m)}~(m\equiv 1[2])$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (-1/2,1);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[above]{$0$};
	\draw (-1/2,0) node[below]{$m$};
	\draw (-1/4,1/2) node[right]{$m$};
	\draw (-3/4,1/2) node[left]{$m$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	\draw (a)--(c);
	
	\fill[fill=white] (c) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
 $\widehat{I_2(m)}~(m\equiv 0[4])$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (-2,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$0$};
	\draw (-3/2,0) node[above]{$m$};
	\draw (-1/2,0) node[above]{$m$};
	
	\draw (a)--(b);
	\draw (a)--(c);
	
	\fill[fill=white] (c) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
 $\widehat{I_2(m)}~(m\equiv 2[4])$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (-2,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$0$};
	\draw (-3/2,0) node[above]{$\frac{m}{2}$};
	\draw (-1/2,0) node[above]{$m$};
	
	\draw (a)--(b);
	\draw (a)--(c);
	
	\fill[fill=white] (c) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
$\widehat{H_3}$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (0,1);
	\coordinate (d) at (-1,1);
	
	\draw (a) node[left]{$1$};
	\draw (b) node[right]{$2$};
	\draw (c) node[right]{$3$};
	\draw (d) node[left]{$0$};
	\draw (-1/2,0) node[below]{$5$};
	\draw (-1/2,1) node[above]{$5$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	\draw (c)--(d);
	\draw (d)--(a);
	
	\fill[fill=white] (d) circle (2.5pt);
	\node[mark size=2.5pt] at (d) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
$\widehat{H_4}$ & \begin{minipage}{0.3\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	\coordinate (d) at (2,0);
	\coordinate (e) at (3,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$3$};
	\draw (d) node[below]{$4$};
	\draw (e) node[below]{$0$};
	\draw (-1/2,0) node[above]{$5$};
	\draw (5/2,0) node[above]{$5$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	\draw (c)--(d);
	\draw (d)--(e);
	
	\fill[fill=white] (e) circle (2.5pt);
	\node[mark size=2.5pt] at (e) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} \\
\hline
\end{tabular}}
\captionof{table}{Compact hyperbolic extensions of $I_2(m)$, $H_3$ and $H_4$.}
\label{hypextdiags}
\end{table}
\medskip

Moreover, in type $H$, the reflection $r_W$ is the only one for which the resulting group $\widehat{W}$ is compact hyperbolic.
\end{prop-def}
\begin{proof}
The expression we give for $r_W$ indicates that $r_W$ indeed is a reflection of $W$. As $r_W$ has order 2, the matrix $\widehat{M}:=(\widehat{m}_{i,j})_{0\le i,j\le n}$ defined by
\[\forall i,j\ge 1,~\widehat{m}_{i,j}=m_{i,j},~\widehat{m}_{0,i}=\widehat{m}_{i,0}:=o(r_Ws_i),~\widehat{m}_{0,0}:=1\]
is indeed a Coxeter matrix and $\widehat{W}$ is the associated Coxeter group. Moreover, we may compute the integers $o(r_Ws_i)$ directly and find the above Coxeter graphs and these are indeed graphs of compact hyperbolic groups, as all those graphs are well-known, see \cite[Appendice]{chein}.

The second statement comes from a tedious, but elementary verification on the 15 (resp. 60) reflections of $H_3$ (resp. $H_4$): only the reflection $r_W$ from the statement gives a graph which appears in the table of \cite{chein}.
\end{proof}
\bigskip
\begin{rem}
A (non-crystallographic) root system $\Phi$ may be associated to $W$. More precisely, $\Phi$ is the orbit under $W$ of the vectors $\alpha_i$ spanning $V$. Then $\Phi$ forms a (non-Euclidean) root system in $V$, which is non-crystallographic in the sense that the condition $\left<\alpha^\vee,\beta\right>\in{\mathbb Z}$ does no longer hold. We still may choose a highest root in $\Phi$. If $W\ne H_3$, then the reflection associated to this highest root is indeed $r_W$.

The extension of $H_3$ with $r_W$ the highest reflection has been considered in \cite{patera-twarock}. It has the following Coxeter graph
\begin{center}
\begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	\coordinate (z) at (0,1);
	
	\draw (-1/2,0) node[above]{$5$};
	\draw (1/2,1/2) node[left]{$5$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	\draw (b)--(z);
	
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (z) circle (2.5pt);
\end{tikzpicture}
\end{center}

However, the sub-graph 
\begin{tikzpicture}[scale=0.6]
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	
	\draw (-1/2,0) node[above]{$5$};
	\draw (1/2,0) node[above]{$5$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}
is of negative type, hence this extension is neither affine or hyperbolic and the sequel does not apply.
\end{rem}
Using the very definitions of $W$ and $\widehat{W}$ as finitely presented groups, we obtain the following result:
\begin{cor}\label{projfromWhattoW}
For any $W$, the assignment
\[\left\{\begin{array}{ccc}
\widehat{s}_0 & \longmapsto & r_W \\ \widehat{s}_i & \longmapsto & s_i\end{array}\right.\]
extends (uniquely) to a surjective reflection-preserving group homomorphism
\[\begin{array}{ccc}\widehat{W}\stackrel{\tiny{\pi}}\longtwoheadrightarrow W.\end{array}\]

Moreover, if $r_W=s_{i_1}\cdots s_{i_k}$ is a reduced expression of $r_W$, then the element $\widehat{r_W}=\widehat{s}_{i_1}\cdots\widehat{s}_{i_k}\in\widehat{W}$ is well-defined and we have
\[\ker\pi=\left<(\widehat{s}_0\widehat{r_W})^{\widehat{W}}\right>,\]
that is, $\ker(\pi)$ is the normal closure of $\widehat{s}_0\widehat{r_W}$ in $\widehat{W}$.
\end{cor}
\begin{proof}
In every reduced expression as in the statement, we have $i_j\ge 1$ so that the element $\widehat{r_W}=\widehat{s}_{i_1}\cdots\widehat{s}_{i_k}$ is in the parabolic subgroup $\widehat{W}_{\{1,\dotsc,n\}}\simeq W$ and thus $\widehat{r_W}$ doesn't depend on the chosen reduced expression for $r_W$. 

Because $\pi$ sends a simple reflection of $\widehat{W}$ to a reflection of $W$, it is clear that is sends any reflection to a reflection.

We have $\pi(\widehat{s}_0\widehat{r_W})=r_W^2=1$ so that the subgroup $N:=\left<(\widehat{s}_0\widehat{r_W})^{\widehat{W}}\right>$ is certainly contained in $\ker(\pi)$. Furthermore, we easily find a presentation of $\widehat{W}/N$ by adding the relation $\widehat{s}_0=\widehat{s}_{i_1}\cdots\widehat{s}_{i_k}$ for $r_W=s_{i_1}\cdots s_{i_k}$ as above to the already known relations for $\widehat{W}$. The composite
\[\left<\overline{s}_0,\overline{s}_1,\dotsc,\overline{s}_n~|~\forall i,j\ge1,~(\overline{s}_i\overline{s}_j)^{m_{i,j}}=1,~\overline{s}_0=\overline{s}_{i_1}\cdots\overline{s}_{i_k}\right>\simeq\widehat{W}/N \longtwoheadrightarrow \widehat{W}/\ker\pi=W\]
maps $\overline{s}_i$ to $s_i$ and is an isomorphism. In particular, this yields an isomorphism of $\widehat{W}$-sets
\[\widehat{W}/N \simeq \widehat{W}/{\ker\pi},\]
forcing $\ker(\pi)$ and $N$ to be conjugate in $\widehat{W}$, hence equal.
\end{proof}
\begin{definition}\label{defofQ}
We denote the kernel of the projection from the previous Corollary by
\[Q:=\ker\pi=\left<(\widehat{s}_0\widehat{r_W})^{\widehat{W}}\right>.\]
\end{definition}
\begin{cor}
With the notation of the above theorem, we have
\[\widehat{W}=Q\rtimes W.\]
\end{cor}
\begin{proof}
The map $s_i\mapsto\widehat{s}_i$ ($i\ge1$) extends to a splitting $W\to\widehat{W}$ of $\pi$.
\end{proof}

\begin{rem}\label{Qisnotabelian}
Let $\Phi$ denote the (non-crystallographic) root system of $W$ and $\widetilde{\alpha}\in\Phi^+$ be the (positive) root associated to the reflection $r_W$, i.e. such that $r_W=s_{\widetilde{\alpha}^\vee}$. If $W\ne H_3$, then $\widetilde{\alpha}=\alpha_0$ is the highest root of $\Phi$. Denote by $\mathrm{t}_{\widetilde{\alpha}^\vee}$ the translation by $\widetilde{\alpha}^\vee$ and by $\sigma^* : W\to GL(V^*)$ the dual of the geometric representation of $W$. We can define a homomorphism
\[\widehat{W} \stackrel{\tiny{\mathbf{a}}}\longrightarrow \mathrm{Aff}(V^*)\]
by sending $\widehat{s}_i$ to $\sigma^*(s_i)$ for $i\ge 1$ and $\mathbf{a}(\widehat{s}_0):=\mathrm{t}_{\widetilde{\alpha}^\vee}\sigma^*(r_W)$. If $W$ is a Weyl group, then $\mathbf{a}$ is injective and identifies $\widehat{W}$ with $W_\mathrm{a}\le\mathrm{Aff}(V^*)$. Moreover, in this case we have
\[Q\simeq \mathbf{a}(Q)=\mathbf{a}\left(\left<(\widehat{s}_0\widehat{s_{\alpha_0^\vee}})^{\widehat{W}}\right>\right)=\left<(\mathbf{a}(\widehat{s}_0)\mathbf{a}(\widehat{s_{\alpha_0^\vee}}))^{\mathbf{a}(\widehat{W})}\right>\]
\[=\left<(t_{\alpha_0^\vee})^{W_\mathrm{a}}\right>=\left<t_{\alpha^\vee},~\alpha\in\Phi\right>\simeq{\mathbb Z}\Phi^\vee\stackrel{\tiny{\text{df}}}=Q^\vee\simeq{\mathbb Z}^n.\]
This is the coroot lattice of $\Phi$ and in particular, the group $Q$ is abelian.

However, a relatively recent result (\cite[Corollary 1.6]{qi_thesis}) states that an irreducible, infinite Coxeter group is affine if and only if it contains an abelian subgroup of finite index and, as $[\widehat{W}:Q]=|W|<\infty$, the group $Q$ cannot be abelian in the hyperbolic case.

Moreover, in the non-crystallographic case, the image of $\mathbf{a}$ is no longer discrete because ${\mathbb Z}\Phi^\vee\subset V^*$ is dense in $V^*$ and also, the morphism $\mathbf{a}$ has no reason to be injective, because we cannot relate the length function on $\widehat{W}$ with separating reflection hyperplanes in $V$ any longer.

The morality is that we should take the geometry of $\widehat{W}$ into account, which is not affine but hyperbolic in the non-crystallographic case.
\end{rem}









\subsection{A key property of the subgroup $Q$}
\hfill


The following result will be crucial in the sequel.
\begin{lem}\label{Qinterpara=1}
The normal subgroup $Q$ trivially intersects every proper parabolic subgroup of $\widehat{W}$, i.e.
\[\forall I\subsetneq \widehat{S},~Q\cap\widehat{W}_I=1.\]
\end{lem}
\begin{proof}
This is clear in the crystallographic case because $Q\simeq {\mathbb Z}\Phi^\vee\simeq{\mathbb Z}^n$.

Recall the morphism $\pi : \widehat{W}\twoheadrightarrow W$. The statement may be rephrased as follows:
\[\forall s\in\widehat{S},~\ker\left(\widehat{W}_{\widehat{S}\setminus\{s\}} \stackrel{\tiny{\pi}}\longrightarrow W\right)=1.\]
For $s=\widehat{s}_0$, this is obvious since $\widehat{W}_{\widehat{S}\setminus\{\widehat{s}_0\}}\stackrel{\tiny{\pi}}\longrightarrow W$ is an isomorphism.

Let $s\in\widehat{S}\setminus\{\widehat{s}_0\}$. Since $\widehat{W}$ is compact hyperbolic, the parabolic subgroup $\widehat{W}_{\widehat{S}\setminus\{s\}}$ is finite. Hence, to prove that the morphism
\[\widehat{W}_{\widehat{S}\setminus\{s\}}\stackrel{\tiny{\pi}}\longtwoheadrightarrow \pi\left(\widehat{W}_{\widehat{S}\setminus\{s\}}\right)\]
is injective, it suffices to prove that 
\begin{equation}\label{bydyer}
\left|\pi\left(\widehat{W}_{\widehat{S}\setminus\{s\}}\right)\right|=\left|\widehat{W}_{\widehat{S}\setminus\{s\}}\right|. \tag{$\star_s$}
\end{equation}
The right-hand side is easily computed using the Coxeter diagram of $\widehat{W}$ (see Table \ref{hypextdiags}). To compute the left-hand side, we proceed by a case-by-case analysis. For $H_4$, we will need the following trick:

Denote by
\[R:=\bigcup_{w\in W}wSw^{-1}=\bigcup_{w\in W}S^w\]
the set of reflections of $W$ and
\[\forall w\in W,~N(w):=\{r\in R~;~\ell(rw)<\ell(w)\}.\]
If $H\le W$ is a reflection subgroup of $W$ (i.e. if $H=\left<H\cap R\right>$), then the set
\[D(H):=\{r\in R~;~N(r)\cap H=\{r\}\}\]
is a set of Coxeter generators of $H$ (see \cite[Theorem 3.3]{dyer90}). In our situation, we find the Coxeter generators $D(\pi(\widehat{W}_{\widehat{S}\setminus\{s\}}))$ and determine the resulting Coxeter diagram, giving the order of $\pi(\widehat{W}_{\widehat{S}\setminus\{s\}})$.
\begin{enumerate}[label=$\bullet$]
\item $W=I_2(m)$ with $m=2k+1$. We have defined $r_W=(s_1s_2)^ks_1$ and we readily compute $s_2={s_1}^{r_W}$ and $s_1={s_2}^{r_W}$ so that $\pi(\widehat{W}_{\widehat{s}_0,\widehat{s}_1})=\pi(\widehat{W}_{\widehat{s}_0,\widehat{s}_2})=W$. On the other hand, we get from the diagram $|\widehat{W}_{\widehat{s}_0,\widehat{s}_1}|=|\widehat{W}_{\widehat{s}_0,\widehat{s}_1}|=2m=|W|$. This proves (\ref{bydyer}) for $s=\widehat{s}_1,\widehat{s}_2$.
\item $W=I_2(m)$ with $m=4k$. In this case we have $r_W=(s_1s_2)^{2k-1}s_1$ and since $s_2=(s_1r_W)^{2k-1}s_1$, we also have $\left<r_W,s_1\right>=W$ and (\ref{bydyer}) is thus true for $s=\widehat{s}_2$ as $\widehat{W}_{\widehat{s}_0,\widehat{s}_1}\simeq W$. Because $s_2r_W=r_Ws_2$, we have $\left<s_2,r_W\right>=A_1\times A_1$ and $\widehat{W}_{\widehat{s}_0,\widehat{s}_2}\simeq A_1\times A_1$ so (\ref{bydyer}) also holds for $s=\widehat{s}_2$.
\item $W=I_2(m)$ with $m=4k+2$. Here, $r_W=(s_1s_2)^{2k}s_1$ and we compute $r_Ws_1r_W=(s_1s_2)^{4k}s_1=s_2s_1s_2=s_1^{s_2}$. In the same way, we get $(s_1(s_1^{s_2}))^ks_1=(s_1s_2)^{2k}s_1=r_W$. This implies $\left<s_1,r_W\right>=\left<s_1,s_1^{s_2}\right>\simeq I_2(2k+1)\simeq \widehat{W}_{\widehat{s}_0,\widehat{s}_1}$. In fact, we have $D(\left<s_1,r_W\right>)=\{s_1,s_1^{s_2}\}$. Now, as above we have $s_2r_W=r_Ws_2$ and $\widehat{W}_{\widehat{s}_0,\widehat{s}_2}\simeq A_1\times A_1\simeq\left<s_2,r_W\right>$.
\item $W=H_3$. Special relations among reflections occur in this case. Namely
\[r_W=s_3^{(s_2s_1)^2},~s_3=r_W^{(s_1s_2)^2},~s_2=s_3(r_Ws_3s_1)^2r_Ws_3,~s_1=(r_Ws_3s_2)^2r_Ws_3r_W.\]
Hence, for $s\in\widehat{S}$, we have $\pi\left(\widehat{W}_{\widehat{S}\setminus\{s\}}\right)=W\simeq\widehat{W}_{\widehat{S}\setminus\{s\}},$ this last isomorphism being given by the diagram of $\widehat{H_3}$. Therefore, all the relations (\ref{bydyer}) hold in this case.
\item $W=H_4$. The additional reflection is
\[r_W=s_4^{(s_3s_2s_1s_2s_3(s_1s_2)^2s_3s_4)^2}.\]
We notice the following relation
\[s_1=s_2s_3(s_4r_W)^2(s_3s_4r_Ws_2(s_3s_4r_W)^2s_2)^3s_3s_4r_Ws_4s_3s_2.\]
This proves that $s_1\in\left<r_W,s_2,s_3,s_4\right>$ so $\pi(\widehat{W}_{\widehat{s}_0,\widehat{s}_2,\widehat{s}_3,\widehat{s}_4})=W\simeq\widehat{W}_{\widehat{s}_0,\widehat{s}_2,\widehat{s}_3,\widehat{s}_4}$. We treat the remaining cases by determining the Dyer generators of the reflection subgroups. Calculations can be done on the sixty reflections of $H_4$ (though easier using \cite{GAP4}). We obtain
\[\pi\left(\widehat{W}_{\widehat{s}_0,\widehat{s}_1,\widehat{s}_3,\widehat{s}_4}\right)=\left<r_W,s_1,s_3,s_4\right>=\left<s_1^{s_2s_3(s_1s_2)^2},s_1,s_3,s_4\right>\simeq A_1\times H_3\simeq\widehat{W}_{\widehat{s}_0,\widehat{s}_1,\widehat{s}_3,\widehat{s}_4}\]
and
\[\pi\left(\widehat{W}_{\widehat{s}_0,\widehat{s}_1,\widehat{s}_2,\widehat{s}_4}\right)=\left<r_W,s_1,s_2,s_4\right>=\left<s_3^{s_4s_2s_1s_2s_3(s_1s_2)^2s_3},s_1,s_2,s_4\right>\simeq I_2(5)^2\simeq\widehat{W}_{\widehat{s}_0,\widehat{s}_1,\widehat{s}_2,\widehat{s}_4}\]
and finally,
\[\pi\left(\widehat{W}_{\widehat{s}_0,\widehat{s}_1,\widehat{s}_2,\widehat{s}_3}\right)=\left<r_W,s_1,s_2,s_3\right>\simeq H_3\times A_1\simeq\widehat{W}_{\widehat{s}_0,\widehat{s}_1,\widehat{s}_2,\widehat{s}_3}.\]
This establishes the relations (\ref{bydyer}) for $W=H_4$, finishing the proof.
\end{enumerate}
\end{proof}
\begin{cor}\label{Qistorsionfree}
The group $Q$ is torsion-free.
\end{cor}
\begin{proof}
Let $q\in Q$ be of finite order. By a theorem of Tits (see \cite[Theorem 3.10]{qi_thesis}), there are $w\in\widehat{W}$ and $J\subset\widehat{S}$ such that $q\in w\widehat{W}_Jw^{-1}$ and $\widehat{W}_J$ is finite. This last condition implies $J\ne\widehat{S}$ and since $Q$ is normal in $\widehat{W}$, we get $q^w\in Q\cap\widehat{W}_J=1$, so $q=1$.
\end{proof}










\subsection{The hyperbolic torus $\mathbf{T}(W)$ of $W$ and its first properties}
\hfill

Before defining the manifold $\mathbf{T}(W)$, we have to study the action of the subgroup $Q\unlhd\widehat{W}$ on the Tits cone of $\widehat{W}$. Recall some notation: define $\widehat{V}:=\mathrm{span}_{\mathbb R}(\alpha_s,~s\in\widehat{S})$ and the bilinear form $\widehat{B}$ given by
\[\widehat{B}(\alpha_s,\alpha_t)=-\cos\left(\frac{\pi}{\widehat{m}_{s,t}}\right),\]
with $(\widehat{m}_{s,t})_{s,t\in\widehat{S}}$ the Coxeter matrix of $(\widehat{W},\widehat{S})$. As $\widehat{W}$ is hyperbolic, the form $\widehat{B}$ has signature $(n-1,1)$. We also have the geometric representation
\[\widehat{\sigma} : \widehat{W} \longhookrightarrow O(\widehat{V},\widehat{B}).\]
Consider the contragredient representation $\widehat{\sigma}^* : \widehat{W}\hookrightarrow GL(\widehat{V}^*)$ and define $(\alpha_s^\vee)_{s\in\widehat{S}}$ to be the dual basis of $\widehat{V}^*$ associated to $(\alpha_s)_{s\in\widehat{S}}$. We have $\widehat{\sigma}^*(w)={}^t{\widehat{\sigma}(w^{-1})}$, that is
\[\forall s,t\in\widehat{S},~\widehat{\sigma}^*(s)(\alpha_t^\vee)=\alpha_t^\vee-2\delta_{s,t}\widehat{B}(-,\alpha_s).\]
The duality pairing of $\widehat{V}$ is denoted $\left<\cdot,\cdot\right>$ as usual. Let moreover
\[\forall s\in\widehat{S},~H_s:=\{\lambda\in\widehat{V}^*~;~\left<\lambda,\alpha_s\right>=0\}~~\text{and}~~A_s:=\{\lambda\in\widehat{V}^*~;~\left<\lambda,\alpha_s\right>>0\}\]
and consider the respective \emph{fundamental chamber} and \emph{Tits cone}
\[C:=\{\lambda\in\widehat{V}^*~;~\left<\lambda,\alpha_{s}\right>>0,~\forall s\in\widehat{S}\}=\bigcap_{s\in\widehat{S}}A_s~~\text{and}~~X:=\bigcup_{w\in\widehat{W}}w(\overline{C}).\]
This is indeed a convex cone and $\overline{C}$ is a fundamental domain for the action of $\widehat{W}$ on $X$. Finally, denote
\[\forall I\subseteq \widehat{S},~C_I:=\left(\bigcap_{s\in I}H_s\right)\cap\left(\bigcap_{s\notin I}A_s\right)\subset\overline{C},\]
in particular $C_\emptyset=C$, $C_{\widehat{S}}=\{0\}$ and we have $\overline{C}=\bigsqcup_{I\subseteq\widehat{S}}C_I$.

In this context, we have the \emph{Coxeter complex}
\[\widehat{\Sigma}:=\Sigma(\widehat{W},\widehat{S})=(X\setminus\{0\})/{\mathbb R}_+^*.\]
This is a $\widehat{W}$-pseudomanifold and we have a decomposition
\[\widehat{\Sigma}=\bigcup_{\substack{w\in\widehat{W} \\ I\subset\widehat{S}}}{\mathbb R}_+^*w(\overline{C_I})\]
which is in fact a $\widehat{W}$-triangulation since ${\mathbb R}_+^*w(\overline{C_I})$ may be identified with the standard $(n-|I|)$-simplex: ${\mathbb R}_+^*w(\overline{C_I})\simeq\Delta^{n-|I|}$. Moreover, since $\widehat{W}$ is infinite, $\widehat{\Sigma}$ is contractible and by \cite[III, \S 2, Corollary 3]{brown_buildings}, since every proper parabolic subgroup of $\widehat{W}$ is finite, $\widehat{\Sigma}$ is in fact a smooth $n$-manifold.
\begin{rem}
All the above construction makes sense for any Coxeter group. If the group $\widehat{W}$ is finite, then $\Sigma(\widehat{W})$ is homeomorphic to the $(\mathrm{rk}(\widehat{W})-1)$-sphere.
\end{rem}

We can give a natural simplicial structure to the Coxeter complex (see \cite[Corollary 2.6]{babson-reiner}). Consider the set of parabolic cosets of $\widehat{W}$
\[P(\widehat{W},\widehat{S}):=\{w\widehat{W}_I~;~w\in\widehat{W},~I\subseteq\widehat{S}\}.\]
We partially order this set as follows:
\[w\widehat{W}_I \preceq w'\widehat{W}_J~\stackrel{\tiny{\text{df}}}\Longleftrightarrow~w\widehat{W}_I\supseteq w'\widehat{W}_J.\]
Notice that $w\widehat{W}_I\preceq w'\widehat{W}_J$ implies $w\widehat{W}_I=w'\widehat{W}_I$ and $J\subset I$. We define the simplicial complex $\Delta(\widehat{W},\widehat{S})$ as the nerve of this poset:
\[\Delta(\widehat{W},\widehat{S}):=\mathcal{N}(P(\widehat{W},\widehat{S}),\preceq).\]

If we denote by $\mathcal{P}(\widehat{\Sigma})$ the poset of faces of $\widehat{\Sigma}$ with respect to the triangulation described above. Then we have an isomorphism of posets
\[\begin{array}{ccc}
(P(\widehat{W},\widehat{S}),\preceq) & \stackrel{\tiny{\sim}}\longrightarrow & (\mathcal{P}(\widehat{\Sigma}),\subseteq) \\ w\widehat{W}_I & \longmapsto & {\mathbb R}_+^*w(\overline{C_I})\end{array}\]
and this yields a $\widehat{W}$-equivariant homeomorphism
\[|\Delta(\widehat{W},\widehat{S})|\stackrel{\tiny{\sim}}\longrightarrow \widehat{\Sigma}.\]

Now, recall that an action of a group $G$ on a space $Z$ is said to be \emph{properly discontinuous} (or a \emph{covering space action}, see \cite[\S 1.3]{hatcher}) if every point $z\in Z$ has an open neighbourhood $z\in U\subset Z$ such that if $g\in G$ is such that $gU\cap U\ne\emptyset$, then $g=1$. In other words, such that
\[O_G(U):=\{g\in G~;~g(U)\cap U\ne\emptyset\}=\{1\}.\]
\begin{lem}\label{Qactsasitshould}
The action of the discrete subgroup $\widehat{\sigma}^*(Q)\le GL(\widehat{V}^*)$ on the Coxeter complex $\widehat{\Sigma}$ is free and properly discontinuous.
\end{lem}
\begin{proof}
Of course, we identify the group $\widehat{W}$ with $\widehat{\sigma}^*(\widehat{W})$. Let $\overline{x}\in\widehat{\Sigma}$ (with $x\in X\setminus\{0\}$). First, we prove that $q(\overline{x})\ne \overline{x}$ for $q\in Q\setminus\{1\}$. To say that $q(\overline{x})=\overline{x}$ amounts to say that $q(x)=ax$ for some $a\in{\mathbb R}_+^*$ and we may assume that $x\in\overline{C}\setminus\{0\}$ since $Q\unlhd\widehat{W}$. There is some $I\subsetneq\widehat{S}$ such that $x\in C_I$. Because $C_I$ is a cone, we have $ax\in C_I\cap q(C_I)\ne\emptyset$ and by \cite[V, \S 4, Proposition 5]{bourbaki456}, we obtain $q(C_I)=C_I$ so $q\in \widehat{W}_I\cap Q=1$ by Lemma \ref{Qinterpara=1}.

To prove that the action is properly discontinuous at $\overline{x}$, we have to find an open neighbourhood $U$ of $\overline{x}$ in $\widehat{\Sigma}$ such that, for $1\ne q\in Q$, we have $U\cap  q(U)=\emptyset$, i.e. $O_Q(U)=\{1\}$.

By definition of the topology on the Coxeter complex, it suffices to prove the statement for $X\setminus\{0\}$. 

First, we show that the action of $\widehat{W}$ is \emph{wandering} at $x$, that is we can find an open neighbourhood $A$ of $x$ such that $O_{\widehat{W}}(A)$ is finite. 

We may assume that $x\in\overline{C}\setminus\{0\}$, say $x\in C_I$ with $I\subsetneq\widehat{S}$. Define $A$ to be the interior in $X\setminus\{0\}$ of the subset $\bigcup_{v\in \widehat{W}_I}v(\overline{C})$. We prove that there are only finitely many $w\in\widehat{W}$ such that $A\cap w(A)\ne\emptyset$. Suppose that $w\in O_{\widehat{W}}(A)$ and choose $a\in A$ with $w(a)\in A$. Notice that we have
\[A\subseteq \bigcup_{u\in\widehat{W}_I}u(C)\cup\bigcup_{\substack{v\in\widehat{W}_I \\ s\in I}}v(H_s\cap\partial\overline{C}).\]
Thus, we distinguish four cases:
\begin{enumerate}[label=$\bullet$]
\item As $\widehat{W}$ acts on $X\setminus\bigcup_{s\in\widehat{S}}H_s$, we cannot have $a\in\bigcup_{v,s}v(H_s\cap\partial\overline{C})$ and $w(a)\in\bigcup_u u(C)$.
\item Similarly, we cannot have $a\in\bigcup_u u(C)$ and $w(a)\in\bigcup_{v,s}v(H_s\cap\partial\overline{C})$.
\item Suppose that $a\in\bigcup_v v(C)$ and $w(a)\in\bigcup_v v(C)$, say $a\in u(C)$ and $w(a)\in v(C)$. This implies $u^{-1}(a)\in C$ and $v^{-1}w(a)=v^{-1}wu(u^{-1}(a))\in C$, thus $uv^{-1}w(C)\cap C\ne\emptyset$ and so $w=vu^{-1}\in\widehat{W}_I$ by Tits' lemma.
\item Suppose now that we have $a\in\bigcup_{v,s}v(H_s\cap\partial\overline{C})$ and $w(a)\in\bigcup_{v,s}v(H_s\cap\partial\overline{C})$, say $a\in u(H_s\cap\partial\overline{C})$ and $w(a)\in v(H_t\cap\partial\overline{C})$. This implies $u^{-1}(a)\in\overline{C}$ and $v^{-1}wu(u^{-1}(a))=v^{-1}w(a)\in\overline{C}$ and by \cite[V, \S 4, Proposition 6]{bourbaki456} we get $v^{-1}w(a)=u^{-1}(a)$ and thus $uv^{-1}w\in (\widehat{W})_a=u\widehat{W}_Ju^{-1}$ for some $J\subsetneq\widehat{S}$ (in fact, $J$ is defined by the condition $\widehat{W}_J=(\widehat{W})_{u^{-1}(a)}$). Therefore, we have $w\in v\widehat{W}_Ju^{-1}$.
\end{enumerate}
In any case, we have
\[O_{\widehat{W}}(A)\stackrel{\tiny{\text{df}}}=\{w\in\widehat{W}~;~w(A)\cap A\ne\emptyset\}\subset\bigcup_{\substack{u,v\in\widehat{W}_I \\ J\subsetneq\widehat{S}}}u\widehat{W}_Jv.\]
However, as $\widehat{W}$ is compact, any proper parabolic subgroup is finite and so this last subset is finite and $O_{\widehat{W}}(A)$ is then finite as well.

The rest of the proof is very standard. For each $w\in O_{\widehat{W}}(A)\setminus\widehat{W}_I$ we have $w(x)\ne x$ and we may choose an open subset $A_w$ such that $x\in A_w\subset A$ and $w(A_w)\cap A_w=\emptyset$ and define
\[B:=\bigcap_{w\in O_{\widehat{W}}(A)\setminus\widehat{W}_I}A_w \subset A.\]
Because $O_{\widehat{W}}(A)$ is finite, $B$ is open and let $w'\in O_{\widehat{W}}(B)\subset O_{\widehat{W}}(A)$. We must have $w'\in\widehat{W}_I$ because otherwise, $\emptyset\ne B\cap w'(B)\subset A_{w'}\cap w'(A_{w'})=\emptyset$ and thus $O_{\widehat{W}}(B)\subset\widehat{W}_I$.

Consider the open subset 
\[U:=\bigcap_{w\in\widehat{W}_I}w(B)\subset B.\]
We have $O_{\widehat{W}}(U)\subset O_{\widehat{W}}(B)\subset\widehat{W}_I$ and $U$ is $\widehat{W}_I$-stable (i.e. $U$ is a $\widehat{W}$-\emph{slice} at $x$). In particular, if $q\in Q\setminus\{1\}$, then $q\notin\widehat{W}_I$ by Lemma \ref{Qinterpara=1} and thus $q\notin O_{\widehat{W}}(U)$.
\end{proof}
We arrive then to the main result of this section. Remark that the Tits form $\widehat{B}$ induces a Riemannian metric on the Coxeter complex $\widehat{\Sigma}$.

\begin{theo}\label{defsteinbergtorus}
Let $(W,S)$ be a finite irreducible Coxeter group of rank $n$ and $(\widehat{W},\widehat{S})$ be either the affine Weyl group associated to $W$ if $W$ is crystallographic, or the extension constructed above otherwise. Then, the orbit space
\[\mathbf{T}(W):=\widehat{\Sigma}/\widehat{\sigma}^*(Q)\]
is a closed, connected, orientable, compact smooth $W$-manifold of dimension $n$.

If $W$ is a Weyl group, then we have an isometry $\widehat{\Sigma}\simeq{\mathbb R}^n$, $Q\simeq{\mathbb Z}^n$ and $\mathbf{T}(W)\simeq{\mathbb R}^n/{\mathbb Z}^n$ is an $n$-torus. Moreover, $\mathbf{T}(W)$ is $W$-diffeomorphic to  a maximal torus of the simply-connected compact Lie group with root system that of $W$.

Otherwise, the Riemannian manifold $\widehat{\Sigma}$ is isometric to the hyperbolic $n$-space $\mathbb{H}^n$ and $\mathbf{T}(W)\simeq\mathbb{H}^n/Q$ is a hyperbolic $W$-manifold.

Furthermore, the canonical projection yields a covering space
\[Q\longhookrightarrow \widehat{\Sigma} \longtwoheadrightarrow \mathbf{T}(W).\]

Finally, the quotient simplicial complex $\Delta(\widehat{W},\widehat{S})/Q$ is a regular $W$-triangulation of $\mathbf{T}(W)$.
\end{theo}
\begin{proof}
Since $\widehat{\Sigma}$ is a closed smooth manifold and the action $\widehat{\sigma}^*(Q)~\rotatebox[origin=c]{-90}{$\circlearrowright$}~\widehat{\Sigma}$ is properly discontinous by Lemma \ref{Qactsasitshould}, the quotient manifold theorem ensures that $\mathbf{T}(W)$ is indeed a closed smooth manifold and by \cite[Proposition 1.40]{hatcher}, the projection $\widehat{\Sigma}\twoheadrightarrow\mathbf{T}(W)$ is a covering map. Moreover, $\mathbf{T}(W)$ is connected since the Coxeter complex is and, as $\overline{C}/{\mathbb R}_+^*\simeq\overline{C}\cap\mathbb{S}^n$ is a $\widehat{W}$-fundamental domain on the Coxeter complex, its projection onto $\mathbf{T}(W)$ is a $W$-fundamental domain, hence $\mathbf{T}(W)$ is compact ($W$ is finite). Since $Q$ is normally generated by $\widehat{s}_0\widehat{r_W}$ and because $\ell(\widehat{r_W})$ is odd, we have $\varepsilon(\widehat{s}_0\widehat{r_W})=1$ and so $Q\le\ker(\varepsilon)$. This proves that the action of $Q$ on $\widehat{\Sigma}$ preserves the orientation, ensuring the orientability of $\mathbf{T}(W)$. 

The comparison with a torus of a Lie group follows directly from the Remark \ref{Qisnotabelian}.

In the non-crystallographic case, let $v^*\in V^*$ be a normalized eigenvector for the negative eigenvalue of $\widehat{B}$. Then the subset $\mathcal{H}:=\{\lambda\in \widehat{V}^*~;~\widehat{B}(\lambda,\lambda)=-1,~\widehat{B}(v^*,\lambda)<0\}$, together with the metric induced by the restriction of $\widehat{B}$ is a Riemannian manifold isometric to the hyperbolic space $\mathbb{H}^n$. We have $\mathbf{T}(W)=\widehat{\Sigma}/Q\simeq\mathcal{H}/Q\simeq\mathbb{H}^n/Q$ and since $Q$ preserves the form $\widehat{B}$, the manifold $\mathbf{T}(W)$ naturally inherits a hyperbolic Riemannian metric.
\end{proof}
\begin{rem}\label{alreadyseeninDavis}
The manifolds $\mathbf{T}(H_3)$ and $\mathbf{T}(H_4)$ have already been discovered in \cite{zimmermann_hyperbolic} and \cite{davis_hyperbolic}. Zimmermann and Davis construct them by taking the orbit under the action of $Q$ (which is defined slightly differently) of hyperbolic polyhedra. However, our approach has the advantage of being more systematic and to work with any finite Coxeter group. For a more detailed study of the Davis manifold and its invariants, see \cite{ratcliffe-tschantz}.
\end{rem}
Recall that, as $\widehat{W}$ is infinite, the Coxeter complex is contractible.
\begin{cor}\label{T(W)isBQ}
The covering space
\[Q\longhookrightarrow \widehat{\Sigma}\longtwoheadrightarrow\mathbf{T}(W)\]
is a universal principal $Q$-bundle. In particular, $\mathbf{T}(W)$ is a classifying space for $Q$ and an Eilenberg-MacLane space
\[\mathbf{T}(W)\simeq B_Q\simeq K(Q,1).\]
\end{cor}














\subsection{Presentation on the fundamental group of $\mathbf{T}(W)$}
\hfill

In this section, we use Poincar\'e's fundamental polyhedron theorem (see \cite[Theorem 11.2.2]{ratcliffe_hyperbolic}) to derive a presentation of the group $\pi_1(\mathbf{T}(W))\simeq Q\unlhd\widehat{W}$ in the non-crystallographic case. The tessellation $\Delta(\widehat{W},\widehat{S})$ of $\widehat{\Sigma}\simeq\mathbb{H}^n$ easily yields a fundamental polyhedron for $Q$ acting on $\widehat{\Sigma}$. 

Choose $v^*\in V^*$ a normalized eigenvector of the Tits form $\widehat{B}$ for its unique negative eigenvalue and consider the subset
\[\mathcal{H}:=\{\lambda\in V^*~;~\widehat{B}(\lambda,\lambda)=-1,~\widehat{B}(v^*,\lambda)<0\}\subset V^*.\]
As already noted in the proof of Theorem \ref{defsteinbergtorus}, the form $\widehat{B}$ induces a Riemannian metric on $\mathcal{H}$ and we have an isometry $\mathcal{H}\simeq\mathbb{H}^n$, where $\mathbb{H}^n$ is the standard hyperbolic $n$-space. By Remark \ref{hyperfromhum}, the fundamental chamber $C$ is included in the subset $\{\lambda~;~\widehat{B}(\lambda,\lambda)<0\}$, hence we can project the punctured Tits cone $X\setminus\{0\}$ on the sheet $\mathcal{H}$ of the hyperbola $\{\lambda~;~\widehat{B}(\lambda,\lambda)=-1\}$ and we get $\widehat{\Sigma}\simeq X\cap\mathcal{H}$. The $n$-simplex
\[\Delta_0:=\overline{C}/{\mathbb R}_+^*\simeq\overline{C}\cap\mathcal{H}\subset\widehat{\Sigma}.\]

Recall that we have denoted $H_s:=\{\lambda~;~\left<\lambda,\alpha_s\right>=0\}$ for $s\in\widehat{S}$. As the subset $L_0:=\overline{C}\cap\bigcap_{s\ne\widehat{s}_0}H_s$ is a line, its intersection with $\mathcal{H}$ is a vertex of the tessellation $\Delta(\widehat{W},\widehat{S})$ and we may consider its star
\[\Delta:=\mathrm{St}\left(L_0\cap\mathcal{H}\right)\stackrel{\tiny{\text{df}}}=\bigcup_{\substack{\sigma\in F_n(\Delta(\widehat{W},\widehat{S})) \\ L_0\cap\mathcal{H}\subset\sigma}}\sigma=\bigcup_{w\in W}w(\Delta_0).\]

We will describe the generators and relations for $\pi_1(\mathbf{T}(W))$ in terms of side-pairing and cycle relations, as in \cite[\S 6.8]{ratcliffe_hyperbolic}. It is easy to see that the facets of $\Delta$ are the $W$-translates of the facet
\[\sigma_0:=H_{\widehat{s}_0}\cap\Delta\in F_{n-1}(\Delta),\]
in other words, $\partial\Delta=\bigcup_{w}w(\sigma_0)$. By \cite[Theorem 6.8.3]{ratcliffe_hyperbolic}, the group $Q=\pi_1(\mathbf{T}(W))$ is generated by the set
\[\Psi:=\{q\in Q~;~\Delta\cap q\Delta\in F_{n-1}(\Delta)\}.\]

\begin{lem}\label{gensforpi1}
The set $\Psi$ of generators of $Q$ is given by the $W$-conjugates of the normal generator of $Q$. In other words, if $r_W\in W$ is the chosen reflection and if $q_0:=\widehat{s}_0r_W\in \widehat{W}$ then we have
\[\Psi=\{{}^wq_0,~w\in W\}=\{wqw^{-1},~\overline{w}\in W/C_W(q_0)\}.\]
\end{lem}

\begin{proof}
Let $1\ne q\in Q$ be such that $\Delta\cap q\Delta$ is a facet of $\Delta$, say $w(\sigma_0)$ for some $w\in W$. We have
\[w(\sigma_0)=\Delta\cap q\Delta=\bigcup_{u,v\in W}u(\Delta_0)\cap qv(\Delta_0)=\bigcup_{u,v\in W}u(\Delta_0\cap u^{-1}qv(\Delta_0)).\]
Since any term of the last union is (empty or) a closed simplex, this means that one of them has to be the whole of $w(\sigma_0)$, so we can find $u,v\in W$ such that
\[u^{-1}w(\sigma_0)=\Delta_0\cap u^{-1}qv(\Delta_0).\]
In particular, we have $u^{-1}w(\sigma_0)\subset \Delta_0$ and since any $\widehat{W}$-orbit meets $\Delta_0$ in only one point, this implies that $u^{-1}w(\sigma_0)=\sigma_0$ and so $u^{-1}w\in \widehat{W}_{\sigma_0}=\left<\widehat{s}_0\right>$ but as $u^{-1}w\in W$, this is possible only when $u=w$. Hence we get
\[\sigma_0=\Delta_0\cap u^{-1}qv(\Delta_0).\]
This implies in turn that $u^{-1}qv\in \left<\widehat{s}_0\right>$ and since $q\ne1$, we must have $u^{-1}qv=\widehat{s}_0$, i.e. $q=u\widehat{s}_0v^{-1}$. Finally, because $q\in Q$, applying the projection $\pi : \widehat{W}\twoheadrightarrow W$ to this equality yields $1=ur_Wv^{-1}$, so $v=ur_W$ and $q=u\widehat{s}_0v^{-1}=uq_0u^{-1}$.
\end{proof}


We can formulate the side-pairing and cycle relations using the combinatorics of $W$. To to this, we need a technical lemma on the centralizer of $q_0$.

\begin{lem}\label{centralizer}
The centralizer of $q_0=\widehat{s}_0r_W$ in $W$ is given by
\[C_W(q_0)=C_W(\widehat{s}_0)=\left<s\in S~;~s\widehat{s}_0=\widehat{s}_0s\right>.\]
In particular, this is (standard) parabolic.
\end{lem}

\begin{proof}
First, we borrow an argument due to Sebastian Schoennenbeck\footnote{\tiny{\url{https://mathoverflow.net/questions/200433/centralizers-of-reflections-in-special-subgroups-of-coxeter-groups}}} to prove the second equality above. Let $w=s_{i_1}\cdots s_{i_r}$ be a reduced expression of an element $w\in C_W(\widehat{s}_0)$. To show that $w$ is in the parabolic subgroup of the statement, since the elements of $C_W(\widehat{s}_0)$ of length 1 are the simple reflections of $W$ commuting with $\widehat{s}_0$, by induction it is enough to show that $\widehat{s}_0s_{i_r}=s_{i_r}\widehat{s}_0$. We have $\ell(w\widehat{s}_0)=\ell(w)+1$ and $\ell(w\widehat{s}_0w^{-1})=\ell(\widehat{s}_0)=1$, so $\ell(w\widehat{s}_0s_{i_r})=\ell(w\widehat{s}_0w^{-1}ws_{i_r})\le1+\ell(ws_{i_r})=\ell(w)$ and thus $\ell(w\widehat{s}_0s_{i_r})=\ell(w)$. Thus, by the exchange condition, there is a reduced expression $w\widehat{s}_0=s_{j_1}\cdots s_{j_r}s_{i_r}$ for $w\widehat{s}_0$ and since $s_{i_1}\cdots s_{i_r}\widehat{s}_0$ is already a reduced expression, by Matsumoto's lemma, there is a finite series of braid-moves from the second to the first. The expression $s_{i_1}\cdots s_{i_r}\widehat{s}_0$ satisfies the property
\begin{equation}\label{redexp}\tag{$\ast$}
\begin{array}{ll}
\text{The expression contains only one occurrence of $\widehat{s}_0$ and there is no simple reflection} \\ 
\text{appearing on the right of $\widehat{s}_0$ that does not commute with it}.\end{array}
\end{equation}
Consider a braid relation $sts\cdots=tst\cdots$ connecting the two expressions of $w\widehat{s}_0$, with $m$ factors on each side and suppose that we apply it to a reduced expression of $w\widehat{s}_0$ verifying (\ref{redexp}). If $s,t\ne\widehat{s}_0$, then the resulting expression still satisfies (\ref{redexp}). Now, if $s=\widehat{s}_0$ say, then $t$ has to commute with $\widehat{s}_0$. Indeed, if not, then the left-hand side of the braid relation contains at least two occurrences of $\widehat{s}_0$ (one on each side of $t$) and, in the right-hand side there is at least one occurrence of $t$ on the right of $\widehat{s}_0$, but none of these occur in the considered reduced expression. Therefore, the reduced expression resulting of the application of the braid move still verifies (\ref{redexp}). In particular, the expression $s_{j_1}\cdots s_{j_r}s_{i_r}$ satisfies (\ref{redexp}) and thus, every simple reflection appearing on the right of $\widehat{s}_0$ must commute with it. In particular, this is the case of $s_{i_r}$, as required.

We now prove that $C_W(q_0)=C_W(\widehat{s}_0)$. Let $w=s_{i_1}\cdots s_{i_r}$ be a reduced expression of an element $w\in C_W(q_0)$. Since $wq_0=q_0w$, we get $\widehat{s}_0w\widehat{s}_0=r_Wwr_W\in W$. Let $\widehat{s}_0w\widehat{s}_0=s_{j_1}\cdots j_{j_k}$ be a reduced expression in $W$. Since $\ell(w\widehat{s}_0)=\ell(w)+1$, we have $\ell(\widehat{s}_0w\widehat{s}_0)\in\{\ell(w),\ell(w)+2\}$. But taking length in the equality $\widehat{s}_0s_{i_1}\cdots s_{i_r}=s_{j_1}\cdots s_{j_k}\widehat{s}_0$ gives $k=r$, that is $\ell(\widehat{s}_0w\widehat{s})=\ell(w)$. In particular, $\ell(\widehat{s}_0w\widehat{s}_0)<\ell(w\widehat{s}_0)$ and by the exchange condition, there is a reduced expression $\widehat{s}_0w\widehat{s}_0=s_{i_1}\cdots \widehat{s_{i_l}}\cdots s_{i_r}\widehat{s}_0$ and since this last expression is in $W$, we must have $s_{i_l}=\widehat{s}_0$ (the reflection $s_{i_l}$ is omitted), thus $\widehat{s}_0w\widehat{s}_0=s_{i_1}\cdots s_{i_r}=w$ and $w\in C_W(\widehat{s}_0)$. The reverse inclusion can be directly checked case by case using the parabolic description of $C_W(\widehat{s}_0)$.
\end{proof}

\begin{rem}
From the diagrams of the hyperbolic extensions we get therefore
\[C_{I_2(2g+1)}(q_0)=1,~C_{I_2(4g+2)}(q_0)=\left<s_2\right>,~C_{I_2(4g)}(q_0)=\left<s_2\right>,\]
\[C_{H_3}(q_0)=\left<s_2\right>,~C_{H_4}(q_0)=\left<s_1,s_2,s_3\right>\simeq H_3.\]
\end{rem}


\begin{theo}\label{presentationQ}
Let $W$ be non-crystallographic and $U:=\{w\in W~;~\ell(ws)>\ell(w),~\forall s\in S~;~s\widehat{s_0}=\widehat{s}_0s\}\approx W/C_W(q_0)$ be the set of minimal length coset representatives modulo the parabolic subgroup $C_W(q_0)$ of $W$. The transitive action of $W$ on $W/C_W(q_0)$ induces an action of $W$ on $U$. Then the fundamental group $\pi_1(\mathbf{T}(W))\simeq Q$ admits the following presentation
\[\pi_1(\mathbf{T}(W))=\left<q_u,~u\in U~|~R_{\text{side}}\cup R_{\text{cycle}}\right>,\]
where 
\[R_{\text{side}}=\{q_uq_v,~u,v\in U~;~u^{-1}vr_W\in C_W(\widehat{s}_0)\}\]
and
\[\begin{array}{c}
R_{\text{cycle}}=\bigg\{q_{w(1)}q_{w(u_2)}\cdots q_{w(u_r)},~w\in W,~u_0,u_1,\dotsc,u_r,u_{r+1}\in U~\text{such that}~u_0=u_{r+1}=1\\
\text{and, for}~i>0,~\left.\left<\widehat{s}_0,\widehat{s}_0^{u_{i+1}^{-1}u_ir_W}\right>~\text{and}~\left<\widehat{s}_0,\widehat{s}_0^{r_Wu_{i-1}^{-1}u_i}\right>~\text{are conjugate under}~C_W(\widehat{s}_0)\right\}.\end{array}\]
\end{theo}

\begin{proof}
Drop the presentation notation and, for $u\in U$, denote $q_u:={}^uq_0=uq_0u^{-1}$, $\sigma_u:=u(\sigma_0)=\Delta\cap q_u(\Delta)$  and $\sigma_u':=q_u^{-1}(\sigma_0)=uq_0(\sigma_0)$. To say that for some $u,u'\in U$ we have $q_uq_{u'}=1$ amounts to say that ${}^uq_0={}^{u'}q_0^{-1}={}^{u'r_W}q_0$, i.e. $u^{-1}u'r_W\in C_W(q_0)$.

For the cycle relations, we follow the method detailed in \cite[\S 6.8]{ratcliffe_hyperbolic}. First notice that each facet of $\sigma_0$ is of the form $\sigma_u$ for some $u\in U$ (see \cite[Theorem 6.7.5]{ratcliffe_hyperbolic}). Choose $\sigma\in F_{n-1}(\Delta)$ and $\tau\in F_{n-2}(\sigma)\subset F_{n-2}(\Delta)$. Recursively define a sequence of facets $\{\sigma_{u_j}\}_{j\in{\mathbb N}^*}$ as follows
\begin{enumerate}[label=$\bullet$]
\item let $\sigma_{u_1}:=\sigma$,
\item let $\sigma_{u_2}$ be the facet of $\sigma$ adjacent to $\sigma_{u_1}':=q_{u_1}^{-1}(\sigma)$ such that $q_{u_1}(\sigma_{u_1}'\cap\sigma_{u_2})=\tau$,
\item for $i>1$, let $\sigma_{u_{i+1}}\in F_{n-2}(\sigma)$ be the facet adjacent to $\sigma_{u_i}':=q_{u_i}^{-1}(\sigma_{u_i})$ such that $q_{u_i}(\sigma_{u_i}'\cap\sigma_{u_{i+1}})=\sigma_{u_{i-1}}'\cap\sigma_{u_i}$.
\end{enumerate}
By \cite[Theorem 6.8.7]{ratcliffe_hyperbolic}, there exists a least integer $k\in{\mathbb N}^*$ such that $\sigma_{u_{i+k}}=\sigma_{u_i}$ for all $i$ and we have $q_{u_1}\cdots q_{u_k}=1$. Moreover, by the Poincar\'e theorem \cite[Theorem 11.2.2]{ratcliffe_hyperbolic}, the set of all such relations (for $\sigma\in F_{n-1}(\Delta)$ and $\tau\in F_{n-2}(\sigma)$), together with the side-pairing relations described above, form a complete set of relations for $Q$.

Let $\sigma\in F_{n-1}(\Delta)\subset W\cdot\sigma_0$. Up to conjugation by an element of $W$, we may assume that $\sigma=\sigma_0$ and so $q_{u_1}=q_0$. Let $u_i,u_j,u_k\in U$ be such that we have some relation
\[q_{u_j}(\sigma_{u_j}'\cap\sigma_{u_k})=\sigma_{u_j}\cap\sigma_{u_i}'\ne\emptyset.\]
We write
\begin{align*}
q_{u_j}(\sigma_{u_j}'\cap\sigma_{u_k})=\sigma_{u_j}\cap\sigma_{u_i}'&~\Longleftrightarrow~\sigma_{u_j}\cap q_{u_j}(\sigma_{u_k})=\sigma_{u_j}\cap q_{u_i}^{-1}(\sigma_{u_i}) \\
&~\Longleftrightarrow~u_j(\sigma_0)\cap q_{u_j}u_k(\sigma_0)=u_j(\sigma_0)\cap q_{u_i}^{-1}u_i(\sigma_0) \\
&~\Longleftrightarrow~u_j(\sigma_0\cap u_j^{-1}q_{u_j}u_k(\sigma_0))=u_j(\sigma_0\cap u_j^{-1}u_iq_0^{-1}(\sigma_0)) \\
&~\Longleftrightarrow~\sigma_0\cap q_0u_j^{-1}u_k(\sigma_0)=\sigma_0\cap u_j^{-1}u_ir_W(\sigma_0),
\end{align*}
and the two hand sides of the last equality are simplices of the tessellation $\Delta(\widehat{W},\widehat{S})$, whose face lattice if the lattice of standard parabolic subgroups of $\widehat{W}$. Hence these two coincides if and only if their stabilizers in $\widehat{W}$ are equal. But this condition depends on the choice of the elements of $U$, but it is straightforward to check that different choices give conjugate stabilizers in $C_W(q_0)$.
\end{proof}

\begin{cor}\label{abelianizationofQ}
The group $\pi_1(\mathbf{T}(H_3))$ (resp. $\mathbf{T}(H_4)$) admits a presentation with 11 (resp. 24) generators , all of whose relations are products of commutators. In particular, we have
\[H_1(\mathbf{T}(H_3),{\mathbb Z})=\pi_1(\mathbf{T}(H_3))^\mathrm{ab}\simeq{\mathbb Z}^{11}~~\text{and}~~H_1(\mathbf{T}(H_4),{\mathbb Z})=\pi_1(\mathbf{T}(H_4))^\mathrm{ab}\simeq{\mathbb Z}^{24}.\]
\end{cor}

\begin{proof}
We apply the above result. For $H_3$, beside the side-pairing relations (which we can immediately simplify by removing half of the $[H_3:C_{H_3}(q_0)]=[H_3:\left<s_2\right>]=60$ generators), we find only one primitive cycle relation (\emph{primitive} meaning starting by $q_0$) of length 3 and one of length 5. Taking the $H_3$-conjugates of these gives 120 relations of length 3 and 120 relations of length 5. But the inverse of each of these relations appears so we can simplify them. We can also remove any cyclic permutation of these relations, which finally yields a presentation for $\pi_1(\mathbf{T}(H_3))$ with 30 generators, 20 relations of length 3 and 12 relations of length 5.

We do the same for $H_4$, where there is only one primitive cycle relation of length 5, which gives a presentation for $\pi_1(\mathbf{T}(H_4))$ with $\tfrac{1}{2}[H_4:C_{H_3}(q_0)]=60$ generators and 144 relations of length 5. 

Using the relations, we can check that some of the generators are superfluous and that the  simplified presentation has the stated number of generators (all among the original generators) and that the relations become trivial, once abelianized. All the formulas are given in Appendix \ref{AB}.
\end{proof}

\begin{rem}
The intermediate presentations of $\pi_1(\mathbf{T}(H_3))$ and $\pi_1(\mathbf{T}(H_4))$ (with 30 generators and 32 relations for $H_3$ and 60 generators and 144 relations for $H_4$) are precisely (up to relabelling) the presentations given in \cite{zimmermann_hyperbolic} and \cite{ratcliffe-tschantz}.
\end{rem}






\subsection{The manifolds $\mathbf{T}(I_2(m))$ as Riemann surfaces}
\hfill


A little bit more can be said about the case of the surfaces $\mathbf{T}(I_2(m))$. Recall that by \cite[Theorem 3.11.1]{jost-riemann}, any Riemannian metric on an oriented 2-manifold $M$ induces a complex structure on $M$ (making $M$ a Riemann surface), called the \emph{conformal structure} induced by the metric.

\begin{cor}\label{T(I2(m))hyperbolicsurface}
Let $g\in{\mathbb N}^*$ and $\mathbf{T}_g$ be either $\mathbf{T}(I_2(2g+1))$, $\mathbf{T}(I_2(4g))$ or $\mathbf{T}(I_2(4g+2))$. Then $\mathbf{T}_g$ is a closed compact Riemann surface of genus $g$. In particular, $\mathbf{T}_g$ is a torus if and only if the corresponding group $I_2$ is crystallographic and we have homeomorphisms
\[\mathbf{T}(I_2(2g+1))\simeq\mathbf{T}(I_2(4g))\simeq\mathbf{T}(I_2(4g+2)).\]
Furthermore, if $g>1$, then $\mathbf{T}_g$ is a hyperbolic surface of finite area $4\pi(g-1)$.
\end{cor}
\begin{proof}
Since the surface $\mathbf{T}_g$ is orientable, the Riemannian metric induced by the one on the Coxeter complex induces a conformal structure on $\mathbf{T}_g$ and the fact that $\mathbf{T}_g$ is hyperbolic if $g>1$ comes from Theorem \ref{defsteinbergtorus}. 

Now, if the Euler characteristic $\chi(\mathbf{T}_g)$ is known, since $\chi(\mathbf{T}_g)=2-2g$ we find the genus of $\mathbf{T}_g$. Because the genus of a closed surface is a compete topological invariant, we obtain the stated homotopy equivalences and the hyperbolic Gauss-Bonnet formula (see \cite[Theorem 9.3.1]{ratcliffe_hyperbolic}) gives $\mathrm{Area}(\mathbf{T}_g)=-2\pi\chi(\mathbf{T}_g)$. Therefore, we only have to compute $\chi(\mathbf{T}_g)$.

Let $m$ be $2g+1$, $4g$ or $4g+2$ according to what $\mathbf{T}_g$ is and 
\[W:=I_2(m)=\left<s,t~|~s^2=t^2=(st)^m=1\right>.\]
We will detail the combinatorics of the $W$-triangulation $\Delta(\widehat{W},\widehat{S})/Q$ in the next section, however we only have to compute $\chi$ and very few information is needed. The rational chain complex associated to the simplicial complex $\Delta(\widehat{W},\widehat{S})$ has the following shape:
\[\xymatrix{{\mathbb Q}[\widehat{W}] \ar[r] & {\mathbb Q}[\widehat{W}/\left<s\right>]\oplus{\mathbb Q}[\widehat{W}/\left<t\right>]\oplus{\mathbb Q}[\widehat{W}/\left<\widehat{s}_0\right>] \ar[r] & {\mathbb Q}[\widehat{W}/\left<s,t\right>]\oplus{\mathbb Q}[\widehat{W}/\left<s,\widehat{s}_0\right>]\oplus{\mathbb Q}[\widehat{W}/\left<t,\widehat{s}_0\right>]}\]
Now, by Lemma \ref{GCWandSDP}, the complex for $\mathbf{T}_g$ is the image of the previous one by the deflation functor $\mathrm{Def}^{\widehat{W}}_W$. Thus, it is of the form
\[\xymatrix{{\mathbb Q}[W] \ar[r] & {\mathbb Q}[W/\left<s\right>]\oplus{\mathbb Q}[W/\left<t\right>]\oplus{\mathbb Q}[W/\left<r\right>] \ar[r] & {\mathbb Q}\oplus{\mathbb Q}[W/\left<s,r\right>]\oplus{\mathbb Q}[W/\left<t,r\right>]},\]
where $r=r_W:=(st)^{\left\lfloor{(m-1)}/{2}\right\rfloor}s\in W$. Therefore the Euler characteristic is given by 
\[\chi(\mathbf{T}_g)=1+[W:\left<s,r\right>]+[W:\left<t,r\right>]-[W:\left<s\right>]-[W:\left<t\right>]-[W:\left<r\right>]+|W|\]
\[=1+[W:\left<s,r\right>]+[W:\left<t,r\right>]-3[W:\left<s\right>]+2m=1-m+[W:\left<s,r\right>]+[W:\left<t,r\right>].\]
Now, we distinguish the three possible cases for $m$ to determine the last two indices. 

If $m=2g+1$ is odd, then we have $r=(st)^gs$, so $s^r=t$ and $t^r=s$ so $\left<s,r\right>=\left<t,r\right>=W$. 

If $m=4g$ is divisible by $4$, then $r=(st)^{2g-1}s$. From the Coxeter diagram of $\widehat{W}=\widehat{I_2(4g)}$, we see that the map interchanging $\widehat{s}_0$ and $t$ and leaving $s$ invariant extends to a non-trivial outer automorphism of $\widehat{I_2(4g)}$ and descends to an outer automorphism of $I_2(4g)$. Taking the image of the relation $r=(st)^{2g-1}s$ under this automorphism yields $t=(sr)^{2g-1}s$ and thus $\left<s,r\right>=W$. Now, since the element $rt=(st)^{2g}$ has order 2, we have $\left<t,r\right>=\{1,t,r,tr\}\simeq C_2\times C_2$.

Now, if $m=4g+2$, then $r=(st)^{2g}s$ and $\left<s,r\right>=\left<s,(st)^{2g}\right>=\left<s,(st)^2\right>=\left<s,tst\right>\simeq I_2(2g+1)$ and because $(rt)^2=(st)^{2g}st(st)^{2g}st=(st)^{4g+2}=1$, we also have $\left<t,r\right>\simeq C_2\times C_2$.

Gathering everything we get
\[[I_2(m):\left<s,r\right>]=\left\{\begin{array}{cc}
2 & \text{if $m=4g+2$}, \\ 1 & \text{otherwise}\end{array}\right.~~\text{and}~~[I_2(m):\left<t,r\right>]=\left\{\begin{array}{cc}
1 & \text{if $m$ is odd}, \\ m/2 & \text{otherwise}\end{array}\right.\]
thus
\[\chi(\mathbf{T}_g)=\left\{\begin{array}{ccc}
3-m & \text{if} & m=2g+1, \\
3-m/2 & \text{if} & m=4g+2, \\
2-m/2 & \text{if} & m=4g,\end{array}\right.\]
in other words,
\[\chi(\mathbf{T}_g)=2-2g.\]
\end{proof}
As the fundamental group of a Riemann surface of genus $g\ge 1$ is well-known (see \cite[\S 1.2]{hatcher}), we obtain a presentation for the group $Q$ in the dihedral case.
\begin{cor}\label{QforI_2}
Let $g\in{\mathbb N}^*$ and $m$ be either $2g+1$, $4g$ or $4g+2$. Let also $\mathbf{T}_g:=\mathbf{T}(I_2(m))$ and $Q$ be the subgroup of $\widehat{I_2(m)}$ constructed in the previous section (see Definition \ref{defofQ}). Then we have
\[Q\simeq\pi_1(\mathbf{T}_g)\simeq\left<x_1,\dotsc,x_g,y_1,\dotsc,y_g~|~[x_1,y_1]\cdots [x_g,y_g]=1\right>\]
and in particular, $Q^\mathrm{ab}\simeq{\mathbb Z}^{2g}$.
\end{cor}

In the cases where $g=1$ that is, if $I_2(m)$ is one of the Weyl groups $I_2(3)=A_2$, $I_2(4)=B_2$ or $I_2(6)=G_2$, then $\mathbf{T}(I_2(m))$ is naturally an \emph{elliptic curve}. More precisely, recalling the notation of the previous section, we have a preferred point 
\[v_0:=\overline{C}\cap\mathcal{H}\cap\bigcap_{s\ne\widehat{s}_0}H_s\in\widehat{\Sigma},\]
and the pair $(\mathbf{T}(I_2(m)),[v_0])$ is a Riemann surface of genus 1 with a marked point, hence an elliptic curve. Notice that, under the isometry $\mathbf{T}(I_2(m))\simeq{\mathbb R}^2/{\mathbb Z}^2$ induced by quotienting the 3-space $\widehat{V}$ by the radical of the Tits form $\widehat{B}_m$ of $\widehat{I_2(m)}=I_2(m)_\mathrm{a}$, the point $[v_0]$ corresponds to the origin.

We can easily identify the elliptic curves $\mathbf{T}(I_2(m))$ (for $m=2,4,6$) in the moduli space $\mathcal{M}_{1,1}\simeq\mathbb{H}^2/PSL_2({\mathbb Z})$ of complex elliptic curves, where $\mathbb{H}^2=\{z\in{\mathbb C}~;~\Im(z)>0\}$ is the Poincar\'e half plane. Recall that to $\tau\in\mathbb{H}^2$ we can associated a $j(\tau)\in{\mathbb C}$ and we have isomorphisms
\[\begin{array}{ccccc}
{\mathbb C} & \stackrel{\tiny{\sim}}\longleftarrow & \mathbb{H}^2/PSL_2({\mathbb Z}) & \stackrel{\tiny{\sim}}\longrightarrow & \mathcal{M}_{1,1} \\ j(\tau) & \longmapsfrom & \tau & \longmapsto & {\mathbb C}/({\mathbb Z}+\tau{\mathbb Z})\end{array}\]
Recall also that $D:=\{z\in\mathbb{H}^2~;~|\Re(z)|\le 1/2,~|z|\ge1\}$ is a fundamental domain for $PSL_2({\mathbb Z})$ acting on $\mathbb{H}^2$. We just have to determine a corresponding element $\tau\in D$ for each case.

For $I_2(3)=A_2$, we have $\mathbf{T}(A_2)\simeq {\mathbb R}\left<\alpha^\vee,\beta^\vee\right>/({\mathbb Z}\alpha^\vee\oplus{\mathbb Z}\beta^\vee)$ and since we have $\left<\alpha^\vee,\beta\right>=\left<\beta^\vee,\alpha\right>=-1$ and $|\alpha|=|\beta|$, the lattice ${\mathbb Z}\alpha^\vee\oplus{\mathbb Z}\alpha^\vee$ is in the $PSL_2({\mathbb Z})$-orbit as ${\mathbb Z}\oplus{\mathbb Z} e^{\frac{2i\pi}{3}}$. Doing the same for $I_2(4)=B_2$, we have $\left<\alpha^\vee,\beta\right>=-1$, $\left<\beta^\vee,\alpha\right>=-2$ and $|\beta|^2=2|\alpha|^2$, so the lattice is ${\mathbb Z}\oplus{\mathbb Z} \sqrt{2}e^{\frac{3i\pi}{4}}\simeq{\mathbb Z}\oplus{\mathbb Z} i$. Finally, for $I_2(6)=G_2$ we find $\left<\alpha^\vee,\beta\right>=-1$, $\left<\beta^\vee,\alpha\right>=-3$ and $|\beta|^2=3|\alpha|^2$, and we obtain the lattice ${\mathbb Z}\oplus{\mathbb Z} \sqrt{3}e^{\frac{5i\pi}{6}}\simeq{\mathbb Z}\oplus{\mathbb Z} e^{\frac{2i\pi}{3}}$.

Hence, for $A_2$ and $G_2$, we find $\tau=e^{\frac{2i\pi}{3}}$, so $j(\tau)=0$ and, for $B_2$, we have $\tau=i$ and $j(\tau)=1728$. In particular, the corresponding curves are defined over ${\mathbb Q}$. We can also take $\tau=e^{\frac{i\pi}{3}}=\left(\begin{smallmatrix}1 & 1 \\ 0 & 1\end{smallmatrix}\right)\cdot e^{\frac{2i\pi}{3}}$ for $A_2$ and the surfaces $\mathbf{T}(I_2(m))$ for $m=2,4,6$ correspond to the three \emph{orbifold points} of $D$, that is, the only three points in $D$ having a non-trivial stabilizer in $PSL_2({\mathbb Z})$. In Weierstrass forms, an equation for $\mathbf{T}(A_2)$ and $\mathbf{T}(G_2)$ is $y^2=x^3-1$ and for $\mathbf{T}(B_2)$, we can take $y^2=x^3-x$.

We now focus on the hyperbolic case where $g>1$. We first notice the following coincidence between the Riemann surface $\mathbf{T}(I_2(m))$.

\begin{prop}\label{isometrybetweenT(I2(m))}
If $g>1$, then we have an isometry (in particular, an isomorphism of Riemann surfaces)
\[\mathbf{T}(I_2(4g+2))\simeq\mathbf{T}(I_2(2g+1))\]
and these two are not isometric to the surface $\mathbf{T}(I_2(4g))$.
\end{prop}

\begin{proof}
Using \cite[Theorem 8.1.5]{ratcliffe_hyperbolic}, it suffices to show that the groups $Q_{2g+1}$ and $Q_{4g+2}$ are conjugate in the positive Lorentz group $PO(1,2)\simeq\mathrm{Isom}(\mathbf{H}^2)\simeq PSL_2({\mathbb R})$ and are not conjugate to $Q_{4g}$.

Let $m:=2g+1$. We first prove that $Q_m$ and $Q_{2m}$ are conjugate in $PO(1,2)$. Denote $I_2(2m)=\left<s,t~|~s^2=t^2=(st)^{2m}=1\right>$ and $\widehat{I_2(2m)}=\left<s,t,\widehat{s}_0\right>$ its hyperbolic extension. Let $s':=s$, $t':=tst=s^t$ and $\widehat{s}_0':=\widehat{s}_0$. Then $\left<s',t',\widehat{s}_0'\right>=\widehat{I_2(m)}$ and $\left<s',t'\right>=I_2(m)$. Recall moreover that we have the reflection $r_{2m}=(st)^{2g}s=((st)^2)^gs=(s't')^gs'=r_m$. Let $\alpha$, $\beta$ and $\gamma$ denote the simple roots of $\widehat{I_2(2m)}$ and $V_{2m}:=\mathrm{span}_{\mathbb R}(\alpha,\beta,\gamma)$. We have the representation
\[\widehat{I_2(2m)}\stackrel{\tiny{\sigma_{2m}}}\longhookrightarrow O(V_{2m},B_{2m}),\]
where
\[B_{2m}=\left(\begin{smallmatrix}1 & -\cos(\pi/2m) & -\cos(\pi/m) \\ -\cos(\pi/2m) & 1 & 0 \\ -\cos(\pi/m) & 0 & 1\end{smallmatrix}\right).\]
In the same way, denote $V_m:=\mathrm{span}_{\mathbb R}(\alpha',\beta',\gamma')$ and $\sigma_m : \widehat{I_2(m)}\longhookrightarrow O(V_m,B_m)$, where
\[B_m=\left(\begin{smallmatrix}1 & -\cos(\pi/m) & -\cos(\pi/m) \\ -\cos(\pi/m) & 1 & -\cos(\pi/m) \\ -\cos(\pi/m) & \cos(\pi/m) & 1\end{smallmatrix}\right).\]
Consider the linear map $P : V_{2m}\to V_m$ with matrix
\[P=\left(\begin{smallmatrix}1 & 1 & 0 \\ 0 & 2\cos(\pi/2m) & 0 \\ 0 & 0 & 1\end{smallmatrix}\right).\]
Then we have $B_m={}^t{P}B_{2m}P$, so $P$ induces an isomorphism $O(V_{2m},B_{2m})\to O(V_m,B_m)$ fitting in a commutative diagram
\[\xymatrix{\widehat{I_2(m)} \ar@{^{(}->}[d] \ar@{^{(}->}^<<<<<{\sigma_m}[r] & O(V_m,B_m) \ar^<<<<<{\sim}[r] & PO(1,2) \\ \widehat{I_2(2m)} \ar@{^{(}->}_<<<<<{\sigma_{2m}}[r] & O(V_{2m},B_{2m}) \ar^{\sim}[u] \ar^<<<<<{\sim}[r] & PO(1,2 \ar^{\sim}[u])}\]
and thus the group $\sigma_m(\widehat{I_2(m)})$ is conjugate in $PO(1,2)$ to a subgroup of $\sigma_{2m}(\widehat{I_2(2m)})$. Therefore, identifying $\widehat{I_2(m)}$ with its image in $\widehat{I_2(2m)}$, it suffices to prove that $Q_{2m}=Q_m$. Recall that $q_{2m}\stackrel{\tiny{\text{df}}}=\widehat{s}_0r_{2m}=\widehat{s}_0r_m=q_m$, so $q_m^{\widehat{I_2(m)}}\subset q_{2m}^{\widehat{I_2(2m)}}$ and thus $Q_m\le Q_{2m}$. Since we have
\begin{align*}
2m[\widehat{I_2(2m)}:\widehat{I_2(m)}]&=[\widehat{I_2(2m)}:\widehat{I_2(m)}][\widehat{I_2(m)}:Q_m]=[\widehat{I_2(2m)}:Q_m] \\
&=[\widehat{I_2(2m)}:Q_{2m}][Q_{2m}:Q_m]=4m[Q_{2m}:Q_m],
\end{align*}
we are left to show that $[\widehat{I_2(2m)}:\widehat{I_2(m)}]=2$. Let $w\in \widehat{I_2(2m)}$. By induction on $\ell(w)$ and because $t$ and $\widehat{s}_0$ commute, we immediately see that $w\in \widehat{I_2(m)}$ if and only if the number of occurrences of $t$ in any reduced expression of $w$ is even. Hence we have $[\widehat{I_2(2m)}:\widehat{I_2(m)}]\le 2$ and since $st$ and $\widehat{s}_0t$ have even order, the map $\widehat{I_2(2m)}\to {\mathbb Z}/2{\mathbb Z}$ sending $s$ and $\widehat{s}_0$ to $0$ and $t$ to $1$ is a homomorphism whose kernel contains $\widehat{I_2(m)}$, hence the result.

We now prove that $Q_{2g+1}$ and $Q_{4g}$ are not conjugate in $PO(1,2)$. It is enough to prove that the elements $\sigma_{2g+1}(q_{2g+1})\in PO(1,2)$ and $\sigma_{4g}(q_{4g})$ have different traces. Write $\widehat{I_2(2g+1)}=\left<s,t,\widehat{s}_0\right>$ and $\widehat{I_2(4g)}=\left<s',t',\widehat{s}_0'\right>$. We have $q_{2g+1}=\widehat{s}_0(st)^gs$ and $q_{4g}=\widehat{s}_0'(st)^{2g}s'$ and we can write explicitly the matrices of the simple reflections in the geometric representation. We diagonalize $st=PdP^{-1}$ and compute $\mathrm{tr}\,(q_{2g+1})=\mathrm{tr}\,(d^gP^{-1}s\widehat{s}_0)$. After calculations, we find
\[\mathrm{tr}\,(q_{2g+1})=8\left(1+\cos\left(\frac{\pi}{2g+1}\right)\right)\cot^2\left(\frac{\pi}{2g+1}\right)-1.\]
Doing the same for $g_{4g}$, we find 
\[\mathrm{tr}\,(q_{4g})=4\cot^2\left(\frac{\pi}{4g}\right)-1.\]
And indeed, we get $\mathrm{tr}\,(q_{2g+1})\ne\mathrm{tr}\,(q_{4g})$ for $g>1$.
\end{proof}

Recall that a \emph{Belyi function} on a Riemann surface $X$ is a holomorphic map $\beta : X \to \widehat{{\mathbb C}}$ which is ramified only over three points of $\widehat{{\mathbb C}}$. Since $\widehat{I_2(m)}$ is a compact triangle group and $Q_m\unlhd \widehat{I_2(m)}$ is torsion-free and of finite index. Thus, by \cite[Theorem 3.10]{jones-wolfart}, the projection
\[\beta : \mathbf{T}(I_2(m))=\mathbb{H}^2/Q_m \longtwoheadrightarrow \mathbb{H}^2/\widehat{I_2(m)}\simeq\widehat{{\mathbb C}}\]
is a Belyi function on $\mathbf{T}(I_2(m))$ of degree $[\widehat{I_2(m)}:Q_m]=2m$. Using \cite[Theorem 1.3]{jones-wolfart}, this implies the following result:

\begin{prop}\label{definableoverQbar}
For any $m\ge 3$, the Riemann surface $\mathbf{T}(I_2(m))$ may be defined over a number field (or equivalently, may be defined over $\overline{{\mathbb Q}}$). Moreover, if $m=5$ or $m\ge 7$, then the 1-skeleton of the tessellation $\Delta(\widehat{I_2(m)})/Q_m$ defines a dessin d'enfant on $\mathbf{T}(I_2(m))$.
\end{prop}

\begin{rem}
It is a reasonable to expect that $\mathbf{T}(I_2(m))$ is definable over ${\mathbb Q}(\cos(2\pi/m))$. This is coherent with the isomorphism $\mathbf{T}(2g+1)\simeq\mathbf{T}(4g+2)$ and with the vertices of the tessellation of $\mathbb{H}^2$, whose coordinates may be chosen in this field. However, we haven't found a proof of this yet. 

This would give a geometric interpretation of the splitting field of a dihedral group.
\end{rem}



\begin{exemple}\label{I2(5,7)}
The triangulation $\Delta(\widehat{I_2(5)})$ is the classical tessellation $\{3,10\}$ of the Poincar\'e disk. More precisely, the Tits form $\widehat{B}$ is given by
\[\widehat{B}=\begin{pmatrix}1 & -c & -c \\ -c & 1 & -c \\ -c & -c & 1\end{pmatrix}~~\text{with}~~c=\cos(\pi/5)\]
and, if $v^*\in V^*$ is a normalized eigenvector for the unique negative eigenvalue of $\widehat{B}$, then we have an identification with the hyperbolic plane
\[\mathcal{H}:=\{\lambda\in V^*~;~\widehat{B}(\lambda,\lambda)=-1,~\widehat{B}(v^*,\lambda)<0\}\simeq\mathbb{H}^2\]
and the stereographic projection on the hyperplane $\widehat{B}(v^*,-)=0$ with pole $\lambda_0$ gives the Poincar\'e disk model for $\mathbb{H}^2$. Under this representation we represent the tessellation $\Delta(\widehat{I_2(5)})=\{3,10\}$ of $\mathcal{H}$ as in Figure \ref{3,10tessellation}, where the black triangles are the images of the fundamental triangle $\overline{C}/{\mathbb R}^*_+\simeq \overline{C}\cap\mathcal{H}$ under elements of odd length.

\begin{center}
\begin{figure}[h!]
\begin{subfigure}[h]{0.3\textwidth}
\centering
\includegraphics[scale=0.245]{I25.png}
\caption{The $\{3,10\}$-tessellation of $\Sigma(\widehat{I_2(5)})\simeq\mathbb{H}^2$.}
\label{3,10tessellation}
\end{subfigure}
\hspace{3cm}
\begin{subfigure}[h]{0.3\textwidth}
\centering
\includegraphics[scale=0.25]{I27.png}
\caption{The $\{3,14\}$-tessellation of $\Sigma(\widehat{I_2(7)})\simeq\mathbb{H}^2$.}
\label{I2(7)}
\end{subfigure}
\caption{Two regular tessellations of the Poincar\'e disk.}
\end{figure}
\end{center}

In this tessellation, we can identify the triangles that are in the $Q$-orbit of $\overline{C}\cap\mathcal{H}$. These are displayed in green in Figure \ref{patreon}. Collapsing these triangles in one gives the surface $\mathbf{T}(I_2(5))$.

\begin{figure}[h!]
\includegraphics[scale=0.185]{TI25patreongreen.png}
\caption{The green triangles form the $Q$-orbit of the fundamental triangle $\overline{C}\cap\mathcal{H}$ inside the Poincar\'e disk.}
\label{patreon}
\end{figure}

We remark that we can extract a fundamental domain for $Q$ on $\mathbf{T}(I_2(5))$ as the projection of the domain displayed in Figure \ref{fundT(I2(5))}.

Rearranging the figure we obtain the triangulation displayed in the Figure \ref{2toruspatreon}, where the points with the same name (resp. the edges with the same color) are identified. We notice that the resulting space is indeed a closed surface of genus 2.

The case of $I_2(m)$ for $m$ odd is pretty similar and we obtain the $\{3,2m\}$-tessellation of the Poincar\'e disk. For instance, the Figure \ref{I2(7)} shows the case of $I_2(7)$.
\begin{center}
\begin{figure}
\begin{subfigure}[h]{0.3\textwidth}
\centering
\includegraphics[scale=0.25]{fundTI25.png}
\caption{Fundamental domain for $Q$ in the Poincar\'e disk.}
\label{fundT(I2(5))}
\end{subfigure}
\hspace{3cm}
\begin{subfigure}[h]{0.3\textwidth}
\centering
\begin{tikzpicture}[scale=2.1]
	\coordinate (a) at (1,0);
	\coordinate (b) at (0.866,0.5);
	\coordinate (c) at (0.5,0.866);
	\coordinate (d) at (0,1);
	\coordinate (e) at (-0.5,0.866);
	\coordinate (f) at (-0.866,0.5);
	\coordinate (g) at (-1,0);
	\coordinate (h) at (-0.866,-0.5);
	\coordinate (i) at (-0.5,-0.866);
	\coordinate (j) at (0,-1);
	\coordinate (k) at (0.5,-0.866);
	\coordinate (l) at (0.866,-0.5);
	
	\draw (a) node[right]{$a$};
	\draw (b) node[right]{$b$};
	\draw (c) node[above right]{$c$};
	\draw (d) node[above]{$b$};
	\draw (e) node[above left]{$a$};
	\draw (f) node[left]{$c$};
	\draw (g) node[left]{$b$};
	\draw (h) node[left]{$c$};
	\draw (i) node[below left]{$a$};
	\draw (j) node[below]{$b$};
	\draw (k) node[below right]{$a$};
	\draw (l) node[right]{$c$};
		
	\draw[red] (a)--(b)
	(j)--(k);
	\draw[green] (b)--(c)
	(f)--(g);
	\draw[blue] (c)--(d)
	(g)--(h);
	\draw[orange] (d)--(e)
	(i)--(j);
	\draw (e)--(f)
	(l)--(a);
	\draw[violet] (h)--(i)
	(k)--(l);
	
	\draw[opacity=0.3] (e)--(b)
	(e)--(c)
	(e)--(g)
	(e)--(h)
	(e)--(j)
	(e)--(l)
	(l)--(b)
	(j)--(h)
	(j)--(l);
	
	\fill[fill=black] (a) circle (1pt);
	\fill[fill=black] (b) circle (1pt);
	\fill[fill=black] (c) circle (1pt);
	\fill[fill=black] (d) circle (1pt);
	\fill[fill=black] (e) circle (1pt);
	\fill[fill=black] (f) circle (1pt);
	\fill[fill=black] (g) circle (1pt);
	\fill[fill=black] (h) circle (1pt);
	\fill[fill=black] (i) circle (1pt);
	\fill[fill=black] (j) circle (1pt);
	\fill[fill=black] (k) circle (1pt);
	\fill[fill=black] (l) circle (1pt);
\end{tikzpicture}
\caption{Fundamental domain for $I_2(5)$ in $\mathbf{T}(I_2(5))=\mathbf{T}_2$.}
\label{2toruspatreon}
\end{subfigure}
\caption{Fundamental domain for $Q$ and its image in $\mathbf{T}(I_2(5))$.}
\end{figure}
\end{center}
\end{exemple}















\section{Equivariant chain complex of $\mathbf{T}(W)$ and computation of homology}



\subsection{The $W$-dg-algebra of $\mathbf{T}(W)$}
\hfill

The combinatorics of the complex $C^\mathrm{cell}_*(\mathbf{T}(W),W;{\mathbb Z})$ is fairly similar to the one of the complex $C^\mathrm{cell}_*(T,W;{\mathbb Z})$ we constructed in the first part and the proofs given above can be applied \emph{verbatim} to this new situation. We obtain the following results:
\begin{theo}\label{chaincomplexforT(W)}
The quotient simplicial complex $\Delta(\widehat{W},\widehat{S})/Q$ is a regular $W$-equivariant triangulation of the manifold $\mathbf{T}(W)$. Recalling the projection $\pi : \widehat{W}\longtwoheadrightarrow W$, the resulting homology ${\mathbb Z}[W]$-chain complex $C_*^\mathrm{cell}(\mathbf{T}(W),W;{\mathbb Z})$ is given (in homogeneous degrees $k$ and $k-1$) by
\[\xymatrix{\cdots \ar[r] & \displaystyle{\bigoplus_{\substack{I\subset \widehat{S} \\ |I|=n-k}}{\mathbb Z}[\pi(\widehat{W}^I)]} \ar^<<<<<{\partial_k}[r] & \displaystyle{\bigoplus_{\substack{I\subset \widehat{S} \\ |I|=n-k+1}}{\mathbb Z}[\pi(\widehat{W}^I)]} \ar[r] & \cdots}\]
with boundaries defined as follows:
\[\forall k\in{\mathbb N},~\forall I\subset\widehat{S}~;~\widehat{S}\setminus I=\{{j_1}<\cdot<{j_{k+1}}\},~(\partial_k)_{|{\mathbb Z}[\pi(\widehat{W}^I)]}=\sum_{u=1}^{k+1}(-1)^up^I_{I\cup\{j_u\}},\]
where, for $J\subset I$, $p^J_I$ denotes the projection
\[p^J_I : \pi(\widehat{W}^J)=\pi(\widehat{W}/\widehat{W}_J) \longtwoheadrightarrow \pi(\widehat{W}/\widehat{W}_I)=\pi(\widehat{W}^I).\]
\end{theo}
\begin{cor}\label{cochaincomplexforT(W)}
The ${\mathbb Z}[W]$-dg-algebra $C^*_{\mathrm{cell}}(\mathbf{T}(W),W;{\mathbb Z})$ associated to the $W$-triangulation $\Delta(\widehat{W},\widehat{S})/Q$ of $\mathbf{T}(W)$ has homogeneous components
\[C^k_\mathrm{cell}(\mathbf{T}(W),W;{\mathbb Z})=\bigoplus_{\substack{I\subset \widehat{S} \\ |I|=n-k}}{\mathbb Z}[\pi({}^I{\widehat{W}})]\simeq\bigoplus_{\substack{I\subset \widehat{S} \\|I|=n-k}}{\mathbb Z}[\pi((\widehat{W})_I)\backslash \widehat{W}],\]
differentials
\[\forall I\subset \widehat{S}~;~\widehat{S}\setminus I=:\{j_0<\cdots<j_k\},~d^k(\pi({}^I{w}))=\sum_{\substack{0\le u \le k+1 \\ j_{u-1}<j<j_u}}(-1)^u\pi\left(\epsilon^{I\setminus\{j\}}_Iw\right),~\epsilon_I^J=\sum_{x\in{}^J_I{\widehat{W}}}x\]
and product 
\[C^p_\mathrm{cell}(\mathbf{T}(W),W;{\mathbb Z})\otimes_{\mathbb Z} C^q_\mathrm{cell}(\mathbf{T}(W),W;{\mathbb Z}) \stackrel{\tiny{\cup}}\longrightarrow C^{p+q}_\mathrm{cell}(\mathbf{T}(W),W;{\mathbb Z}),\]
induced by a product
\[{\mathbb Z}[{}^I\widehat{W}]\otimes_{\mathbb Z}\Z[{}^J\widehat{W}] \stackrel{\tiny{\cup}}\longrightarrow {\mathbb Z}[{}^{I\cap J}\widehat{W}],\]
given by the formula
\[\forall x,y\in\widehat{W},~{}^I{x}\cup{}^J{y}=\delta_{\max(I^\complement),\min(J^\complement)}\times\left\{\begin{array}{cc}
{}^{I\cap J}((xy^{-1})_Jy) & \text{if }xy^{-1}\in\widehat{W}_I\widehat{W}_J \\ 0 & \text{otherwise}.\end{array}\right.\]
\end{cor}
\begin{rem}
We make several observations on the previous results.
\begin{enumerate}[label=$\bullet$]
\item As explained in \cite[\S 2.3]{babson-reiner}, a quotient simplicial complex of the form $\Delta(\widehat{W},\widehat{S})/H$ (with $H\le\widehat{W}$) has a an interpretation in terms of double cosets. In our case, we have an isomorphism of posets
\[\begin{array}{ccc}
\left(\mathcal{P}(\Delta(\widehat{W},\widehat{S})/Q),\subseteq\right) & \stackrel{\tiny{\sim}}\longrightarrow & \left(\{(I,Qw\widehat{W}_I)\}_{I\subseteq\widehat{S},~w\in\widehat{W}},\preceq\right) \\ \pi(\widehat{W}^I) \ni \pi(w^I) & \longmapsto & (I,Qw\widehat{W}_I) \end{array}\]
where the order $\preceq$ on the second factor is defined by
\[(I,Qw\widehat{W}_I)\preceq(J,Qw'\widehat{W}_J)~\stackrel{\tiny{\text{df}}}\Longleftrightarrow~\left\{\begin{array}{c}I\supseteq J \\ Qw\widehat{W}_I\supseteq Qw'\widehat{W}_J\end{array}\right.\]
and we may rephrase the above results using this poset.
\item As the subgroup $Q$ is torsion-free, it contains no reflection and we have $Q\le\ker(\varepsilon)$, so by the general result \cite[Proposition 2.4.2]{reiner_complexes}, the quotient $\mathbf{T}(W)$ is an orientable pseudomanifold. The Theorem \ref{defsteinbergtorus} can be seen as a refinement of this result in our particular setting.
\item The triangulation $\Delta(\widehat{H_3})/Q$ (resp. $\Delta(\widehat{H_4})/Q$) has f-vector $(4,124,240,120)$ (resp. $(266,7920,29280,36000,14400)$). In particular, the Euler characteristics are given by
\[\chi(\mathbf{T}(H_3))=0~~\text{and}~~\chi(\mathbf{T}(H_4))=26.\]
We notice that the f-vector of our triangulation of $\mathbf{T}(H_4)$ is far bigger than the one found in \cite[\S 3]{ratcliffe-tschantz}, which is $(1,60,144,60,1)$ but of course, this last one doesn't correspond to an equivariant triangulation. Notice finally that the general Gauss-Bonnet formula gives $\mathrm{Vol}(\mathbf{T}(H_4))=\frac{4\pi^2}{3}\chi(\mathbf{T}(H_4))=104\pi^2/3\approx 342.15$.
\end{enumerate}
\end{rem}










\subsection{The homology $W$-representation of $\mathbf{T}(W)$}
\hfill

We can now determine the action of $W$ on $H_*(\mathbf{T}(W),{\mathbb Z})$. In fact, as for the classical tori, we will show that there is no torsion in $H_*(\mathbf{T}(W),{\mathbb Z})$ but we shall decompose the representations $H_*(\mathbf{T}(W),{\mathbb Q}(W))$ over a splitting field ${\mathbb Q}(W)$ of $W$, which is bigger than ${\mathbb Q}$ in the non-crystallographic cases.

Recall from \cite[Theorem 5.3.8]{geck-pfeiffer} that a splitting field for $W$ is given by
\[{\mathbb Q}(W)={\mathbb Q}(\cos(2\pi/m_{s,t}),~s,t\in S)={\mathbb Q}(\chi_\sigma(w),~w\in W)\subset{\mathbb R},\]
where $\chi_\sigma=\mathrm{tr}(\sigma)$ is the character of the geometric representation $\sigma : W\to GL(V)$ of $W$. If $W$ is a Weyl group, then ${\mathbb Q}(W)={\mathbb Q}$ and we have
\[{\mathbb Q}(I_2(m))={\mathbb Q}(\cos(2\pi/m))~~\text{and}~~{\mathbb Q}(H_3)={\mathbb Q}(H_4)={\mathbb Q}(\sqrt{5}).\]

We suppose from now on that $W$ is one of the groups $H_3$, $H_4$ or $I_2(m)$, with $m\ge 3$ and we keep the notation of the previous section. The first groups to be determined are the top and bottom homology of $\mathbf{T}(W)$. Recall that we have $n=\mathrm{rk}(W)=\dim\mathbf{T}(W)$.
\begin{prop}\label{H0andHn}
Let $\mathds{1}$ and $\varepsilon$ respectively the trivial and signature modules over ${\mathbb Z}[W]$. Then we have isomorphisms of ${\mathbb Z}[W]$-modules
\[\left\{\begin{array}{ccc}
H_0(\mathbf{T}(W),{\mathbb Z})\simeq\mathds{1}, \\
H_n(\mathbf{T}(W),{\mathbb Z})\simeq\varepsilon.\end{array}\right.\]
\end{prop}
\begin{proof}
Since $\widehat{\Sigma}$ is path-connected, its quotient $\mathbf{T}(W)$ is path-connected too and is orientable by Theorem \ref{defsteinbergtorus}. Thus, we have an isomorphism of abelian groups
\[H_0(\mathbf{T}(W),{\mathbb Z})\simeq{\mathbb Z}\simeq H_n(\mathbf{T}(W),{\mathbb Z}).\]
It is clear that $H_0(\mathbf{T}(W),{\mathbb Z})$ is the trivial module and, as ${\mathbb Z}[W]$-modules we have $H_n(\mathbf{T}(W),{\mathbb Z})=\ker(\partial_n)$ with
\[\begin{array}{ccccc}
\partial_n & : & {\mathbb Z}[W] & \longrightarrow & \bigoplus_{i=0}^n{\mathbb Z}[W/\left<s_i\right>] \\ & & w & \longmapsto & \sum_{i=0}^n(-1)^iw\left<s_i\right>\end{array}\]
where $s_i=\pi(\widehat{s}_i)$ is a simple reflection of $W$ for $i\ge 1$ and $s_0:=r_W=\pi(\widehat{s}_0)$. Define $e:=\sum_w\varepsilon(w)w\in{\mathbb Z}[W]$ with $\varepsilon(w)=(-1)^{\ell(w)}$ and notice that $we=\varepsilon(w)e$ for $w \in W$ and $\partial_n(e)=0$. Let $x=\sum_w x_w w\in{\mathbb Z}[W]$ such that $\partial_n(x)=0$. Then, for all $0\le i \le n$, we have $\sum_wx_ww\left<s_i\right>=0$. Fixing $1\le i \le n$, we can choose a set $\{w_1,\dotsc,w_k\}$ of representatives of the left coset $W/\left<s_i\right>$ (the minimal length representatives for instance). We have
\[0=\sum_{w\in W} x_ww\left<s_i\right>=\sum_{j=1}^k x_{w_j}w_j\left<s_i\right>+\sum_{j=1}^kx_{w_js_i}w_js_i\left<s_i\right>=\sum_j(x_{w_j}+x_{w_js_i})w_j\left<s_i\right>,\]
hence $x_{w_j}+x_{w_js_i}=0$ for all $1\le j \le k$. This implies $x_w+x_{ws_i}=0$ for all $w\in W$ and doing this for every $i\ge 1$ gives $x_w+x_{ws}=0$ for all $w\in W$ and $s\in S$, in other words, $x_w=\varepsilon(w)x_1$ for $w\in W$ and $x=x_1e\in{\mathbb Z} e$.
\end{proof}


\iffalse
Now we prove that $H_*(\mathbf{T}(W),{\mathbb Z})$ has no torsion and determine the rank of the $H_1$.
\begin{lem}\label{Htorsionfreeandrank}
The homology $H_*(\mathbf{T}(W),{\mathbb Z})$ is torsion-free and we have
\[H_1(\mathbf{T}(H_3),{\mathbb Z})\simeq{\mathbb Z}^{11}~~\text{and}~~H_1(\mathbf{T}(H_4),{\mathbb Z})\simeq{\mathbb Z}^{24}.\]
\end{lem}
\begin{proof}
For simplicity, if $A$ is an abelian group, we denote by $\mathrm{Tors}(A)$ its torsion subgroup.

We also denote $H_i:=H_i(\mathbf{T}(W),{\mathbb Z})$ and $H^i:=H^i(\mathbf{T}(W),{\mathbb Z})$. By Poincar\'e duality and the universal coefficients theorem, we have
\[\mathrm{Tors}(H^{n-i})=\mathrm{Tors}(H_i)=\mathrm{Tors}(H^{i+1}).\]

First, the case of $W=I_2(m)$ is obvious since $\mathbf{T}(I_2(m))$ is an orientable closed surface.

Let now $W=H_3$. We have $\mathrm{Tors}(H_i)=\mathrm{Tors}(H_i)=0$ for $i=0,3$. Moreover, $\mathrm{Tors}(H_2)=\mathrm{Tors}(H^3)=0$. Thus, it suffices to prove that $\mathrm{Tors}(H_1)=0$ and to compute its rank. But since $\mathbf{T}(W)\simeq B_Q$, we have
\[H_1(\mathbf{T}(W),{\mathbb Z})=H_1(B_Q,{\mathbb Z})\simeq H_1(Q,{\mathbb Z})=Q^\mathrm{ab}.\]
Therefore, we have to prove that $Q^\mathrm{ab}$ is torsion-free and to compute its rank. This can be readily done using the command \texttt{AbelianInvariantsNormalClosureFpGroup} of \cite{GAP4}. This involves an abelianized Reidemeister-Schreier method (see \cite[II, \S 4]{lyndon-schupp}) to compute a presentation of the group $Q^\mathrm{ab}$. Technically, these calculations can be done by hand but are really tedious, especially in comparison to the simple following commands:

\begin{center}
\begin{minipage}{1.1\textwidth}
\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single]
!gapprompt@gap>| !gapinput@F := FreeGroup( "a", "b", "c", "z" );; AssignGeneratorVariables( F );;|
!gapprompt@gap>| !gapinput@W := F / ParseRelators( [ a, b, c, z ], "a2 = b2 = c2 = z2 = |
!gapprompt@>| !gapinput@(ab)5 = (bc)3 = (cz)5 = (za)3 = (ac)2 = (bz)2 = 1" );;|
!gapprompt@gap>| !gapinput@rW := W.3 ^ ( ( W.2 * W.1 ) ^ 2 );;|
!gapprompt@gap>| !gapinput@Collected( AbelianInvariantsNormalClosureFpGroup( W, Subgroup( W, [ W.4 * rW ] ) ) );|
[ [ 0, 11 ] ]
\end{Verbatim}
\end{minipage}
\end{center}
and thus $Q^\mathrm{ab}\simeq{\mathbb Z}^{11}$.

Finally, if $W=H_4$, we have $\mathrm{Tors}(H_i)=\mathrm{Tors}(H^i)=0$ for $i=0,4$ and thus $\mathrm{Tors}(H_3)=\mathrm{Tors}(H^4)=0$. Moreover, as above we have $H_1\simeq Q^\mathrm{ab}$ and $\mathrm{Tors}(H_2)=\mathrm{Tors}(H^3)=\mathrm{Tors}(H_1)=\mathrm{Tors}(Q^\mathrm{ab})$. Therefore, as for $H_3$, we prove that $Q^\mathrm{ab}\simeq {\mathbb Z}^{24}$ using GAP.
\iffalse
\begin{center}
\begin{minipage}{1.1\textwidth}
\begin{Verbatim}[commandchars=!@|,fontsize=\small,frame=single]
!gapprompt@gap>| !gapinput@F := FreeGroup( "a", "b", "c", "d", "z" );; AssignGeneratorVariables( F );;|
!gapprompt@gap>| !gapinput@W := F / ParseRelators( [ a, b, c, d, z ], "a2 = b2 = c2 = d2 = z2 = |
!gapprompt@>| !gapinput@(ab)^5 = (bc)3 = (cd)3 = (dz)5 = (ac)2 = (ad)2 = (az)2 = (bd)2 = (bz)2 = (cz)2 = 1" );;|
!gapprompt@gap>| !gapinput@rW := W.4 ^ ( ( ( W.1 ^ ( W.2 * W.3 ) ) * ( ( W.1 * W.2 ) ^ 2 ) * W.3 * W.4 ) ^ 2 );;|
!gapprompt@gap>| !gapinput@Collected( AbelianInvariantsNormalClosureFpGroup( W, Subgroup( W, [ W.5 * rW ] ) ) );|
[ [ 0, 24 ] ]
\end{Verbatim}
\end{minipage}
\end{center}
\fi
\end{proof}
\begin{rem}
The polyhedral approach of \cite{zimmermann_hyperbolic} and \cite{davis_hyperbolic} has the advantage of allowing to determine a presentation of $Q$, using the Poincar\'e fundamental polyhedron theorem \cite[Theorem 11.2.2]{ratcliffe_hyperbolic}. From this presentation, we can theoretically derive a presentation of $Q^\mathrm{ab}$ and thus obtain the previous result avoiding the use of the computer. In \cite[Theorem]{zimmermann_hyperbolic}, the presentation of $Q$ is given in type $H_3$. It is rather simple and it is proved there that $Q^\mathrm{ab}\simeq{\mathbb Z}^{11}$. However in type $H_4$, the presentation given in \cite[\S 4]{ratcliffe-tschantz} is not quite handy (it has at least 60 generators and 144 relations of length 5 each) and they still use the computer to determine the homology of $\mathbf{T}(H_4)$.
\end{rem}
\fi


\begin{prop}\label{PD}
The homology $H_*(\mathbf{T}(W),{\mathbb Z})$ is torsion-free and the Poincar\'e duality on $\mathbf{T}(W)$ induces isomorphisms of ${\mathbb Z}[W]$-modules
\[H_{n-i}(\mathbf{T}(W),{\mathbb Z})^\vee\simeq H_i(\mathbf{T}(W),{\mathbb Z})^\vee\otimes_{\mathbb Z}\varepsilon.\]
\end{prop}

\begin{proof}
It suffices to prove that $H_*(\mathbf{T}(W),{\mathbb Z})$ is torsion-free, the Poincar\'e pairing $H^i(\mathbf{T}(W),{\mathbb Z}) \otimes H_{n-i}(\mathbf{T}(W),{\mathbb Z}) \to H_n(\mathbf{T}(W),{\mathbb Z})=\varepsilon$ and the universal coefficient theorem implying the second one.

For simplicity, if $A$ is an abelian group, we denote by $\mathrm{Tors}(A)$ its torsion subgroup.

We also denote $H_i:=H_i(\mathbf{T}(W),{\mathbb Z})$ and $H^i:=H^i(\mathbf{T}(W),{\mathbb Z})$. Again using Poincar\'e duality and the universal coefficients theorem, we get
\[\mathrm{Tors}(H^{n-i})=\mathrm{Tors}(H_i)=\mathrm{Tors}(H^{i+1}).\]

Since $n\le 4$, it remains to show that $\mathrm{Tors}(H_1)=0$. This is always true for an orientable surface and still holds for $H_3$ and $H_4$, by Corollary \ref{abelianizationofQ}.
\end{proof}


The above Lemma, combined with the Hopf trace formula (see below) provides enough information to determine the homology representation of $\mathbf{T}(W)$.
\begin{lem}[{Hopf trace formula, \cite[Chap. 4, \S 7, Theorem 6]{spanier} or \cite[Lemma 2.4]{lin_lefschetz}}]\label{hopf}
Let $\Bbbk$ be a field, $C_\bullet$ be a bounded chain complex of finite dimensional $\Bbbk$-vector spaces and $f\in\mathrm{End}_\Bbbk(C_\bullet)$ be an endomorphism of $C_\bullet$. If $H_\ast(f)\in\mathrm{End}_\Bbbk(H_*(C))$ denotes the induced endomorphism in homology, then we have the formula
\[\sum_i (-1)^i\mathrm{tr}_\Bbbk(f_i)=\sum_i(-1)^i\mathrm{tr}_\Bbbk(H_i(f)).\]
\end{lem}

This can be readily applied to our situation to obtain the following formula. As a reminder, if $G$ is a (discrete) group, $H\le G$ is a subgroup and if $M$ is an $H$-module, we denote by $M{\uparrow}_H^G$ the \emph{induced module} of $M$; it is a $G$-module. Similarly, the \emph{restricted module} of a $G$-module $N$ is denoted $N{\downarrow}_H^G$. Observe that we have a canonical isomorphism of ${\mathbb Q}[G]$-modules ${\mathbb Q}[G/H]\simeq\mathds{1}{\uparrow}_H^G$. Recall also that if $N\unlhd G$ and if $M$ is a $G$-module, then its \emph{deflation} $\mathrm{Def}_{G/N}^G(M)$ is a $G/N$-module.

In our context, we have isomorphisms of ${\mathbb Q}[\widehat{W}]$-modules
\[C_k^\mathrm{cell}(\widehat{\Sigma},\widehat{W};{\mathbb Q})=\bigoplus_{I\subset\widehat{S}~;~|I|=n-k}\mathds{1}{\uparrow}_{\widehat{W}_I}^{\widehat{W}}.\]
Thus 
\[C_k^\mathrm{cell}(\mathbf{T}(W),W;{\mathbb Q})=\mathrm{Def}^{\widehat{W}}_W(C_k^\mathrm{cell}(\widehat{\Sigma},\widehat{W};{\mathbb Q}))=\bigoplus_{I\subset\widehat{S}~;~|I|=n-k}\mathds{1}{\uparrow}_{\pi(\widehat{W}_I)}^W,\]
and we obtain the following result:
\begin{lem}\label{hopfforW}
We have the following equality of virtual rational characters of $W$
\[\sum_{I\subsetneq\widehat{S}}(-1)^{|I|}\mathrm{Def}^{\widehat{W}}_W\left(\mathds{1}{\uparrow}_{\widehat{W}_I}^{\widehat{W}}\right)=\sum_{I\subsetneq\widehat{S}}(-1)^{|I|}\mathds{1}{\uparrow}_{\pi(\widehat{W}_I)}^W=(-1)^n\sum_{i=0}^n(-1)^iH_i(\mathbf{T}(W),{\mathbb Q}).\]
\end{lem}

For notation simplicity, we shall use the conventions of \cite{geck-pfeiffer} to denote the irreducible characters of $W$. We start with $I_2(m)$.
\begin{theo}\label{homologyI2}
Let $m\ge 3$ and $\zeta_m:=\exp({{2i\pi}/{m}})$. Following \cite[\S 5.3.4]{geck-pfeiffer}, for $1\le j \le \lfloor (m-1)/2\rfloor$, we consider the following representation of $I_2(m)=\left<s,t~|~s^2=t^2=(st)^m=1\right>$
\[\widetilde{\rho}_j : I_2(m) \to GL_2({\mathbb C})~~\text{defined by}~~\widetilde{\rho}_j(s):=\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}~~\text{and}~~\widetilde{\rho}_j(st):=\begin{pmatrix}\zeta_m^j & 0 \\ 0 & \zeta_m^{-j}\end{pmatrix}.\]

By \cite[Theorem 5.3.8]{geck-pfeiffer}, the real field $\Bbbk_m:={\mathbb Q}(\cos(2\pi/m))$ is a splitting field for $I_2(m)$ and we choose $\rho_j$ to be a realization of $\widetilde{\rho}_j$ over $\Bbbk_m$.

Then, the first homology representation of $\mathbf{T}(I_2(m))$ is given by
\[H_1(\mathbf{T}(I_2(m)),\Bbbk_m)=\left\{\begin{array}{cc}
\displaystyle{\bigoplus_{1\le j \le (m-1)/2}\rho_j} & \text{if $m$ is odd}, \\[.5em]
\displaystyle{\bigoplus_{\substack{1\le j \le m/2-1 \\ j~\text{odd}}}\rho_{j}} & \text{if $m$ is even}.\end{array}\right.\]
Recall also that $H_0(\mathbf{T}(I_2(m)),{\mathbb Q})=\mathds{1}$ and $H_2(\mathbf{T}(I_2(m)),{\mathbb Q})=\varepsilon$.
\end{theo}
\begin{proof}
We already have obtained the last statement above in Proposition \ref{H0andHn}. For the first homology, we let $\chi_j:=\mathrm{tr}(\rho_j)$ be the character of $\rho_j$ and, denoting by $\mathrm{Reg}_{\Bbbk_m}=\Bbbk_m[I_2(m)]$ the regular module, lemma \ref{hopfforW} yields the following equality of virtual characters of $I_2(m)$
\[H_1(\mathbf{T}(I_2(m)),\Bbbk_m)=\mathds{1}+\varepsilon-\mathrm{Reg}_{\Bbbk_m}-\sum_{\emptyset\ne I\subsetneq\{\widehat{s}_0,s,t\}}(-1)^{|I|}\mathds{1}{\uparrow}_{\pi(\widehat{I_2(m)}_I)}^{I_2(m)}.\]
We deal with each case separately. Recall the computations of the images in $I_2(m)$ of the parabolic subgroups of $\widehat{I_2(m)}$ from the proof of the Corollary \ref{T(I2(m))hyperbolicsurface}.
\begin{enumerate}[label=$\bullet$]
\item $m=2k+1$ is odd. We have $r:=r_W=(st)^ks$. Hence $s^{r}=t$ and $t^{r}=s$ so $I_2(m)=\left<s,r\right>=\left<t,r\right>$. Furthermore, in this case we have (cf \cite[\S 5.3.4]{geck-pfeiffer}) $\mathrm{Reg}_{\Bbbk_m}=\mathds{1}+\varepsilon+\sum_j 2\chi_j$ and the above formula reduces to
\[H_1(\mathbf{T}(I_2(m)),\Bbbk_m)=3\cdot\mathds{1}{\uparrow}_{\left<s\right>}^{I_2(m)}-3\cdot\mathds{1}-\sum_j2\chi_j.\]
Now, by \cite[\S 6.3.5]{geck-pfeiffer} we have $\mathds{1}{\uparrow}_{\left<s\right>}^{I_2(m)}=\mathds{1}+\sum_j\chi_j$ and thus
\[H_1(\mathbf{T}(I_2(m)),\Bbbk_m)=\sum_j\chi_j.\]
\item $m=4k$. In this case we have $r=(st)^{2k-1}s$ and let $a:=st$. The conjugacy classes of $I_2(m)$ are given as follows
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}
\hline
Representative & $1$ & $a$ & $a^2$ & $\cdots$ & $a^{2k}$ & $a^{2k+1}$ & $s$ & $t$ \\
\hline
Cardinality & $1$ & $2$ & $2$ & $\cdots$ & $2$ & $1$ & $2k+1$ & $2k+1$ \\
\hline
\end{tabular}
\end{center}

First, we determine the characters $\mathds{1}{\uparrow}_{\left<x,r\right>}^{I_2(m)}$ for $x=s,t$. In the proof of \ref{T(I2(m))hyperbolicsurface} we have seen that $t=(sr)^{2k-1}s$, so $\left<s,r\right>=I_2(m)$. Next, as detailed in \cite[\S 5.3.4]{geck-pfeiffer}, the character $\chi_j$ is given by
\[\chi_j(a^i)=2\cos\left(\frac{2\pi ij}{m}\right)~~\text{and}~~\chi_j(sa^i)=0.\]
We have $\left<t,r\right>=\{1,t,r,a^{2k}\}\simeq C_2\times C_2$ and by Frobenius reciprocity
\[\forall j,~\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\chi_j\right)_W=\left(\mathds{1},\chi_j{\downarrow}_{\left<t,r\right>}^{I_2(m)}\right)_{\left<t,r\right>}=\frac{\chi_j(1)+\chi_j(t)+\chi_j(r)+\chi_j(a^{2k})}{4}\]
\[=\frac{\chi_j(1)+2\chi_j(t)+\chi_j(a^{2k})}{4}=\frac{1+\cos(\pi j)}{2}=\left\{\begin{array}{cc}
1 & \text{if $j$ is even} \\ 0 & \text{otherwise}.\end{array}\right.\]
The 1-dimensional irreducible representations of $I_2(m)$ other that $\mathds{1}$ and $\varepsilon$ are given by $\varepsilon_s(s)=\varepsilon_t(t)=1$ and $\varepsilon_s(t)=\varepsilon_t(s)=-1$. Therefore, $\mathrm{Reg}_{\Bbbk_m}=\mathds{1}+\varepsilon+\varepsilon_s+\varepsilon_t+\sum_j2\chi_j$. We directly compute using Frobenius reciprocity
\[\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\varepsilon_s\right)_{I_2(m)}=\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\varepsilon\right)_{I_2(m)}=0\]
and
\[\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\varepsilon_t\right)_{I_2(m)}=\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\mathds{1}\right)_{I_2(m)}=1\]
and hence
\[\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)}=\mathds{1}+\varepsilon_t+\sum_{j~\text{even}}\chi_j.\]
On the other hand, by \cite[\S 6.3.5]{geck-pfeiffer}, we have $\mathds{1}{\uparrow}_{\left<s\right>}^{I_2(m)}=\mathds{1}+\varepsilon_s+\sum_j\chi_j$ and $\mathds{1}{\uparrow}_{\left<t\right>}^{I_2(m)}=\mathds{1}+\varepsilon_t+\sum_j\chi_j$. Putting everything together and remembering that $t$ and $r$ are conjugate yields
\[H_1(\mathbf{T}(I_2(m)),\Bbbk_m)=\mathds{1}+\varepsilon-\mathrm{Reg}_{\Bbbk_m}-\sum_{\emptyset\ne I\subsetneq\{\widehat{s}_0,s,t\}}(-1)^{|I|}\mathds{1}{\uparrow}_{\pi(\widehat{I_2(m)}_I)}^{I_2(m)}\]
\[=\mathds{1}+\varepsilon-\mathrm{Reg}_{\Bbbk_m}+\mathds{1}{\uparrow}_{\left<s\right>}^{I_2(m)}+2\cdot\mathds{1}{\uparrow}_{\left<t\right>}^{I_2(m)}-\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)}-2\cdot\mathds{1}=\sum_{j~\text{odd}}\chi_j.\]
\item $m=4k+2$. We proceed in the same way, noticing that $r=(st)^{2k}s=a^{2k}s$. The characters $\mathds{1}{\uparrow}_{\left<s\right>}^{I_2(m)}$ and $\mathds{1}{\uparrow}_{\left<t\right>}^{I_2(m)}$ are determined as above. We compute
\[\left(\mathds{1}{\uparrow}_{\left<s,r\right>}^{I_2(m)},\varepsilon_s\right)_{I_2(m)}=\left(\mathds{1},\varepsilon_s{\downarrow}_{\left<s,r\right>}^{I_2(m)}\right)_{\left<s,r\right>}=1=\left(\mathds{1}{\uparrow}_{\left<s,r\right>}^{I_2(m)},\mathds{1}\right)_{I_2(m)}\]
but since $\deg(\mathds{1}{\uparrow}_{\left<s,r\right>}^{I_2(m)})=[I_2(m):\left<s,r\right>]=[I_2(m):\left<s,a^{2k}\right>]=2$ this implies $\mathds{1}{\uparrow}_{\left<s,r\right>}^{I_2(m)}=\mathds{1}+\varepsilon_s$. Now, we have $\left<t,r\right>=\{1,t,r,a^{2k+1}\}\simeq C_2\times C_2$ and using again the Frobenius reciprocity we obtain
\[\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\chi_j\right)_{I_2(m)}=\left(\mathds{1},\chi_j{\downarrow}_{\left<t,r\right>}^{I_2(m)}\right)_{\left<t,r\right>}=\frac{\chi_j(1)+\chi_j(t)+\chi_j(r)+\chi_j(a^{2k+1})}{4}\]
\[=\frac{\chi_j(1)+\chi_j(a^{2k+1})}{4}=\frac{1+\cos(\pi j)}{2}.\]
Since $\mathds{1}{\downarrow}_{\left<t,r\right>}^{I_2(m)}\ne\mathds{1}\ne\mathds{1}{\downarrow}_{\left<t,r\right>}^{I_2(m)}$ we also get
\[\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\varepsilon_s\right)_{I_2(m)}=\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\varepsilon_t\right)_{I_2(m)}=\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\varepsilon\right)_{I_2(m)}=0\]
and
\[\left(\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)},\mathds{1}\right)_{I_2(m)}=1.\]
Finally,
\[\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)}=\mathds{1}+\sum_{j~\text{even}}\chi_j.\]
As above, we conclude that
\[H_1(\mathbf{T}(I_2(m)),\Bbbk_m)=\varepsilon-\mathrm{Reg}_{\Bbbk_m}+2\cdot\mathds{1}{\uparrow}_{\left<s\right>}^{I_2(m)}+\mathds{1}{\uparrow}_{\left<t\right>}^{I_2(m)}-\mathds{1}{\uparrow}_{\left<s,r\right>}^{I_2(m)}-\mathds{1}{\uparrow}_{\left<t,r\right>}^{I_2(m)}=\sum_{j~\text{odd}}\chi_j,\]
as claimed.
\end{enumerate}
\end{proof}
\begin{theo}\label{homologyH3}
With the notation of \cite[Appendix C, Table C.1]{geck-pfeiffer}, we have
\[\forall 0\le i\le 3,~H_i(\mathbf{T}(H_3),{\mathbb Q}(\sqrt{5}))=\left\{\begin{array}{ccc}
\mathds{1} & \text{if} & i=0, \\[.5em]
3_s'\oplus\overline{3'_s}\oplus5_r & \text{if} & i=1, \\[.5em]
3_s\oplus\overline{3_s}\oplus5'_r & \text{if} & i=2, \\[.5em]
\varepsilon & \text{if} & i=3. \end{array}\right.\]
\end{theo}
\begin{proof}
Consider the virtual character $\chi_H:=\sum_{I\subsetneq\widehat{S}}(-1)^{|I|+1}\mathds{1}{\uparrow}_{\pi(\widehat{H_3}_I)}^{H_3}$. For $\chi\in\mathrm{Irr}(H_3)$, we may compute
\[\left(\chi,\chi\right)_{H_3}=\sum_{I\subsetneq\widehat{S}}(-1)^{|I|+1}\left(\mathds{1}{\uparrow}_{\pi(\widehat{H_3}_I)}^{H_3},\chi_H\right)_{H_3}\]
\[=\sum_{I\subsetneq\widehat{S}}(-1)^{|I|+1}\left(\mathds{1},\chi{\downarrow}_{\pi(\widehat{H_3}_I)}^{H_3}\right)_{\pi(\widehat{H_3}_I)}=\sum_{\substack{I\subsetneq\widehat{S} \\ w\in\pi(\widehat{H_3}_I)}}(-1)^{|I|+1}\chi(w).\]
We obtain
\[\chi_H=\varepsilon-\mathds{1}-3_s-\overline{3_s}+3_s'+\overline{3'_s}+5_r-5_r'\]
and therefore, using lemma \ref{hopfforW}
\[H_2(\mathbf{T}(H_3))-H_1(\mathbf{T}(H_3))=-3_s-\overline{3_s}+3_s'+\overline{3_s'}+5_r-5'_r.\]
But from Lemma \ref{Htorsionfreeandrank}, we have $\dim(H_1(\mathbf{T}(H_3)))=\dim(H_2(\mathbf{T}(H_3)))=11=\dim(3_s+\overline{3_s}+5_r')$, so
\[H_1(\mathbf{T}(H_3),{\mathbb Q}(\sqrt{5}))=3'_s+\overline{3'_s}+5_r~~\text{and}~~H_2(\mathbf{T}(H_3),{\mathbb Q}(\sqrt{5}))=3_s+\overline{3_s}+5_r'=H_1(\mathbf{T}(H_3),{\mathbb Q}(\sqrt{5}))\otimes\varepsilon.\]
\end{proof}

Finally we treat the case of $H_4$.
\begin{theo}\label{homologyH4}
With the notation of \cite[Appendix C, Table C.2]{geck-pfeiffer}, we have
\[\forall 0\le i \le 4,~H_i(\mathbf{T}(H_4),{\mathbb Q}(\sqrt{5}))=\left\{\begin{array}{ccc}
\mathds{1} & \text{if} & i=0, \\[.5em]
4_t\oplus\overline{4_t}\oplus16_r' & \text{if} & i=1, \\[.5em]
6_s\oplus\overline{6_s}\oplus30_s\oplus\overline{30_s} & \text{if} & i=2, \\[.5em]
4_t'\oplus\overline{4_t'}\oplus16_r & \text{if} & i=3, \\[.5em]
\varepsilon & \text{if} & i=4. \end{array}\right.\]
\end{theo}
\begin{proof}
As for the previous proof, we let $\chi_H:=\sum_{I\subsetneq\widehat{S}}(-1)^{|I|}\mathds{1}{\uparrow}_{\pi(\widehat{H_4}_I)}^{H_4}$ and
\[\forall \chi\in\mathrm{Irr}(H_4),~\left(\chi,\chi_H\right)_{H_4}=\sum_{I\subsetneq\widehat{S},~w\in\pi(\widehat{H_4}_I)}(-1)^{|I|}\chi(w).\]
This leads to
\[\chi_H=\mathds{1}+\varepsilon-4_t-\overline{4_t}-4_t'-\overline{4_t'}+6_s+\overline{6_s}-16_r-16'_r+30_s+\overline{30_s}.\]
Since $\dim(H_1(\mathbf{T}(H_4)))=\dim(H_3(\mathbf{T}(H_4)))=24$ we obtain
\[H_2(\mathbf{T}(H_4),{\mathbb Q}(\sqrt{5}))=30_s+\overline{30_s}+6_s+\overline{6_s}\]
and
\[H_1(\mathbf{T}(H_4))+H_3(\mathbf{T}(H_4))=4_t+4_t'+\overline{4_t}+\overline{4_t'}+16_r+16_r'.\]
But since the representations $H_1(\mathbf{T}(H_4))$ and $H_3(\mathbf{T}(H_4))$ must be realizable over ${\mathbb Q}$ and because of the Poincar\'e duality pairing between the two, we are left with the following four possibilities:
\begin{center}
\begin{tabular}{c|c|c|c|c}
$H_1(\mathbf{T}(H_4))$ & $4_t+\overline{4_t}+16_r$ & $4_t'+\overline{4_t'}+16_r$ & $4_t+\overline{4_t}+16_r'$ & $4'_t+\overline{4'_t}+16'_r$ \\
\hline
$H_3(\mathbf{T}(H_4))$ & $4'_t+\overline{4'_t}+16'_r$ & $4_t+\overline{4_t}+16'_r$ & $4'_t+\overline{4'_t}+16_r$ & $4_t+\overline{4_t}+16_r$
\end{tabular}
\end{center}

However, the ${\mathbb Q}[H_4]$-module $H_1(\mathbf{T}(H_4),{\mathbb Q})$ is a sub-quotient of the module
\[C_1^\mathrm{cell}(\mathbf{T}(H_4),H_4;{\mathbb Q})=\sum_{\substack{I\subsetneq\widehat{S} \\ |I|=3}}\mathds{1}{\uparrow}_{\pi(\widehat{H_4}_I)}^{H_4}\]
and we compute
\[\left(C_1^\mathrm{cell}(\mathbf{T}(H_4)),16_r\right)_{H_4}=0~\Longrightarrow~\left(H_1(\mathbf{T}(H_4)),16_r\right)_{H_4}=0.\]
Hence, only $16_r'$ can be a direct factor of $H_1(\mathbf{T}(H_4),{\mathbb Q}(\sqrt{5}))$. In the same fashion we compute
\[\left(C_1^\mathrm{cell}(\mathbf{T}(H_4)),4_t'\right)_{H_4}=0~\Longrightarrow~\left(H_1(\mathbf{T}(H_4)),4_t'\right)_{H_4}=0\]
and thus only the third column of the table above is possible.
\end{proof}
\begin{rem}
In \cite[\S 3]{ratcliffe-tschantz} and \cite[\S 2.2]{martelli_hyperbolic}, the homology of $\mathbf{T}(H_4)$ is also described, but only as a ${\mathbb Z}$-module. 
\end{rem}

Finally, we exhibit another algebraic meaning of the Euler characteristic of $\mathbf{T}(W)$. The Poincar\'e series of $\widehat{I_2(m)}$, $\widehat{H_3}$ and $\widehat{H_4}$ can be found in \cite[\S 3.1, Table 7.4 and Table 7.5]{chapovalov}. Using these expressions, we immediately obtain the following corollary:
\begin{cor}\label{euler+poincare=love}
Let $W$ be a finite irreducible Coxeter group. If $W(q)$ (resp. $\widehat{W}(q)$) denotes the Poincar\'e series of $W$ (resp. of its extension $\widehat{W}$), then the Euler characteristic of $\mathbf{T}(W)$ is given by
\[\chi(\mathbf{T}(W))=\left.\frac{W(q)}{\widehat{W}(q)}\right|_{q=1}\]

Moreover, the geometric representation $\sigma$ of $W$ is always a direct summand of $H_1(\mathbf{T}(W),{\mathbb Q}(W))$ for every $W$ and the two are equal if and only if $W$ is crystallographic. In particular
\[\bigoplus_{\alpha\in\mathrm{Gal}({\mathbb Q}(W)/{\mathbb Q})}\sigma^\alpha~\text{is a direct summand of}~H_1(\mathbf{T}(W),{\mathbb Q}).\]
\end{cor}
\begin{rem}
With \cite{chapovalov} it can be seen that the quotient $W(q)/\widehat{W}(q)$ is a polynomial in $q$, but we cannot hope for a generalization of the Bott factorization theorem as in the affine case, i.e. a formula of the form 
\[\frac{W(q)}{W_\mathrm{a}(q)}=\prod_{i=1}^n1-q^{d_i-1},\]
with $\{d_i\}$ the degrees of $W$. Indeed, the polynomial $H_4(q)/\widehat{H_4}(q)$ is irreducible of degree $60$.
\end{rem}






























\begin{appendix}
\section{Presentations of $\mathbf{T}(H_3)$ and $\mathbf{T}(H_4)$}\label{AB}

In this appendix, we make the presentation of $\pi_1(\mathbf{T}(W))$ from Theorem \ref{presentationQ} more explicit in the cases of $H_3$ and $H_4$.

We write $\Psi_3=:\{f_i,~1\le i \le 30\}$ for the set of (abstract) generators of $\pi_1(\mathbf{T}(H_3))$. We obtain the presentation
\[\pi_1(\mathbf{T}(H_3))=\left<\Psi_3~|~R_3\right>,\]
where
\scriptsize{\[R_3:=\{f_{22}f_{25}^-f_{4}^-,~f_{10}^-f_{12}^-f_{11},~f_{1}f_{25}f_{20},~f_{2}f_{26}f_{21},~f_{1}f_{29}f_{18},~f_{3}f_{28}f_{16},~f_{2}f_{30}f_{19},~f_{3}f_{27}f_{17},~f_{5}f_{26}f_{23}^-,~f_{5}f_{6}^-f_{12},\]

\[f_{15}f_{13}f_{14}^-,~f_{4}f_{7}^-f_{11},~f_{16}f_{14}^-f_{21}^-,~f_{17}f_{13}^-f_{20}^-,~f_{10}f_{8}f_{9}^-,~f_{28}^-f_{9}^-f_{23},~f_{29}f_{24}^-f_{7},~f_{27}^-f_{8}^-f_{22},~f_{6}^-f_{24}f_{30}^-,~f_{15}f_{18}^-f_{19},\]

\[f_{23}f_{19}^-f_{12}f_{21}f_{24}^-,~f_{22}f_{16}^-f_{10}f_{17}f_{23}^-,~f_{28}^-f_{20}f_{21}^-f_{27}f_{15}^-,~f_{1}f_{8}f_{13}^-f_{4}^-f_{3}^-,~f_{2}f_{9}f_{14}^-f_{5}^-f_{3}^-,~f_{4}f_{30}^-f_{10}f_{29}f_{5}^-,\]

\[f_{27}^-f_{12}^-f_{25}f_{9}^-f_{7},~f_{22}f_{18}^-f_{11}f_{20}f_{24}^-,~f_{28}^-f_{11}^-f_{26}f_{8}^-f_{6},~f_{17}f_{19}^-f_{25}f_{14}^-f_{29}^-,~f_{1}f_{6}f_{15}f_{7}^-f_{2}^-,~f_{16}f_{18}^-f_{26}f_{13}^-f_{30}^-\}.\]}

\normalsize{We} verify the following relations among generators:

\scriptsize{\[f_1=f_{13}f_{3}f_{8}^-f_{4},~f_{5}=f_{10}f_{8}f_{3}^-f_{14}^-f_{2},~f_6=f_{11}f_{8}f_{3}^-f_{14}^-f_{2},~f_7=f_{11}f_{4},~f_9=f_{10}f_{8},~f_{12}=f_{11}f_{10}^-,~f_{15}=f_{14}f_{13}^-,\]

\[f_{16}=f_{21}f_{14},~f_{18}=f_{19}f_{14}f_{13}^-,~f_{20}=f_{17}f_{13}^-,~f_{22}=f_{8}f_{3}^-f_{17}^-,~f_{23}=f_{10}f_{8}f_{3}^-f_{14}^-f_{21}^-,~f_{24}=f_{11}f_{8}f_{3}^-f_{14}^-f_{19}^-,\]

\[f_{25}=f_{4}^-f_{8}f_{3}^-f_{17}^-,~f_{26}=f_{2}^-f_{21}^-,~f_{27}=f_{3}^-f_{17}^-,~f_{28}=f_{3}^-f_{14}^-f_{21}^-,~f_{29}=f_{4}^-f_{8}f_{3}^-f_{14}^-f_{19}^-,~f_{30}=f_{2}^-f_{19}^-.\]}

\normalsize{Thus}, replacing the generators on the left hand sides by the words on the right hand side yields  a presentation $\pi_1(\mathbf{T}(H_3))=\left<\Psi_3'~|~R_3'\right>$ with 
\[\Psi_3'=\{f_i,~i\in\underbrace{\{2,3,4,8,10,11,13,14,17,19,21\}}_{=:I_{11}}\},\]

and

\scriptsize{\[R_3':=\{f_{13}f_{3}f_{8}^-f_{4}f_{8}f_{13}^-f_{4}^-f_{3}^-,~f_{2}f_{10}f_{8}f_{14}^-f_{2}^-f_{14}f_{3}f_{8}^-f_{10}^-f_{3}^-,~f_{21}f_{14}f_{3}f_{17}f_{13}^-f_{21}^-f_{3}^-f_{17}^-f_{13}f_{14}^-,~f_{21}f_{14}f_{13}f_{14}^-f_{19}^-f_{2}^-f_{21}^-f_{13}^-f_{19}f_{2},\]

\[f_{8}f_{3}^-f_{17}^-f_{14}^-f_{21}^-f_{10}f_{17}f_{21}f_{14}f_{3}f_{8}^-f_{10}^-,~f_{21}f_{14}f_{3}f_{11}^-f_{2}^-f_{21}^-f_{8}^-f_{11}f_{8}f_{3}^-f_{14}^-f_{2},~f_{17}f_{19}^-f_{4}^-f_{8}f_{3}^-f_{17}^-f_{14}^-f_{19}f_{14}f_{3}f_{8}^-f_{4},\]

\[f_{17}f_{3}f_{10}f_{11}^-f_{4}^-f_{8}f_{3}^-f_{17}^-f_{8}^-f_{10}^-f_{11}f_{4},~f_{4}f_{19}f_{2}f_{10}f_{4}^-f_{8}f_{3}^-f_{14}^-f_{19}^-f_{14}f_{8}^-f_{10}^-f_{2}^-f_{3},~f_{3}f_{4}f_{13}f_{8}^-f_{11}f_{10}^-f_{3}^-f_{2}f_{10}f_{8}f_{13}^-f_{4}^-f_{11}^-f_{2}^-,\]

\[f_{10}f_{8}f_{3}^-f_{14}^-f_{21}^-f_{19}^-f_{11}f_{10}^-f_{21}f_{19}f_{14}f_{3}f_{8}^-f_{11}^-,~f_{8}f_{3}^-f_{17}^-f_{13}f_{14}^-f_{19}^-f_{11}f_{17}f_{13}^-f_{19}f_{14}f_{3}f_{8}^-f_{11}^-\}.\]}

\normalsize{We} notice that any of the above relation becomes trivial once abelianized and indeed, $\pi_1(\mathbf{T}(H_3))^{\mathrm{ab}}={\mathbb Z}^{11}$.

In terms of $H_3$-conjugates of $q_0:=\widehat{s}_0r_{H_3}$, the 11 generators $\{f_i,~i\in I_{11}\}$ of $\pi_1(\mathbf{T}(H_3))$ given above may be written $f_i=q_0^{u_i}$ where the $u_i$'s are given by

\[\{u_i,~i\in I_{11}\}=\{xy(s_2s_1)^2,~y^{s_2s_1}s_2s_1,~s_1,~s_2s_1,~y^{-1}s_3,~s_2y^{-1}s_3,~(x^{-1}s_1)^2,~s_3(x^{-1}s_1)^2,~s_3ys_1,~s_2y,~y\},\]
with
\[x:=s_3s_1^{s_2},~y:=((s_1s_2)^2)^{s_3}.\]

We can do the same for $H_4$. We have a presentation
\[\pi_1(\mathbf{T}(H_4))=\left<\Psi_4~|~R_4\right>,\]
where $\Psi_4=:\{f_i,~1\le i \le 60\}$ and

\newpage

\tiny{\[R_4=\{f_{3}^-f_{9}^-f_{19}^-f_{10}f_{2},~f_{1}f_{7}f_{17}^-f_{5}^-f_{2}^-,~f_{1}f_{8}f_{18}^-f_{6}^-f_{3}^-,~f_{2}f_{10}f_{19}^-f_{9}^-f_{3}^-,~f_{1}f_{13}f_{22}f_{16}^-f_{4}^-,~f_{2}f_{11}f_{20}f_{14}^-f_{4}^-,~f_{3}f_{12}f_{21}f_{15}^-f_{4}^-,~f_{6}f_{25}^-f_{17}^-f_{23}f_{9}^-,\]

\[f_{3}^-f_{58}f_{36}^-f_{29}f_{51},~f_{52}f_{57}^-f_{30}^-f_{49}f_{9},~f_{6}f_{28}^-f_{22}f_{33}f_{12}^-,~f_{9}f_{26}^-f_{20}f_{32}f_{12}^-,~f_{9}^-f_{45}^-f_{35}^-f_{13}^-f_{58}^-,~f_{17}f_{39}^-f_{32}^-f_{33}f_{36},~f_{5}f_{25}^-f_{18}^-f_{24}f_{10}^-,~f_{6}^-f_{28}f_{22}^-f_{33}^-f_{12},\]

\[f_{5}f_{29}^-f_{22}f_{34}f_{11}^-,~f_{10}f_{11}^-f_{32}f_{21}f_{27}^-,~f_{10}f_{27}^-f_{21}f_{32}f_{11}^-,~f_{17}f_{31}^-f_{14}f_{16}^-f_{29},~f_{23}^-f_{7}f_{8}^-f_{24}f_{19}^-,~f_{53}f_{15}f_{50}f_{34}^-f_{59}^-,~f_{23}^-f_{36}^-f_{22}f_{42}f_{26},~f_{1}^-f_{57}f_{43}^-f_{32}f_{54},\]

\[f_{19}f_{27}^-f_{15}f_{14}^-f_{26},~f_{59}^-f_{13}^-f_{36}^-f_{44}^-f_{10}^-,~f_{59}^-f_{34}^-f_{50}f_{15}f_{53},~f_{2}^-f_{60}f_{40}^-f_{30}f_{53},~f_{24}^-f_{35}^-f_{22}f_{41}f_{27},~f_{45}f_{35}f_{13}f_{58}f_{9},~f_{45}f_{28}f_{29}^-f_{44}^-f_{19}^-,~f_{45}f_{42}^-f_{50}f_{21}f_{49}^-,\]

\[f_{9}^-f_{52}^-f_{57}f_{30}f_{49}^-,~f_{8}f_{1}f_{3}^-f_{6}^-f_{18}^-,~f_{51}f_{29}f_{36}^-f_{58}f_{3}^-,~f_{18}f_{40}^-f_{32}^-f_{34}f_{35},~f_{8}f_{57}f_{14}f_{42}f_{35},~f_{3}f_{52}^-f_{31}^-f_{39}f_{60}^-,~f_{17}f_{36}f_{33}f_{32}^-f_{39}^-,~f_{8}f_{24}^-f_{19}f_{23}f_{7}^-,\]

\[f_{2}f_{56}^-f_{41}f_{33}^-f_{54}^-,~f_{8}f_{30}^-f_{21}f_{33}f_{13}^-,~f_{7}f_{31}^-f_{20}f_{34}f_{13}^-,~f_{9}f_{38}f_{41}f_{16}f_{55},~f_{44}f_{38}f_{12}f_{60}f_{5},~f_{18}f_{24}^-f_{10}f_{5}^-f_{25},~f_{44}f_{41}^-f_{50}f_{20}f_{48}^-,~f_{18}f_{30}^-f_{15}f_{16}^-f_{28},\]

\[f_{25}^-f_{39}^-f_{20}f_{42}f_{28},~f_{47}^-f_{36}f_{38}^-f_{49}f_{18}^-,~f_{47}^-f_{13}^-f_{54}^-f_{60}f_{25},~f_{17}f_{5}f_{2}f_{1}^-f_{7}^-,~f_{18}f_{45}^-f_{26}^-f_{31}f_{46},~f_{1}^-f_{13}^-f_{22}^-f_{16}f_{4},~f_{7}f_{57}f_{15}f_{41}f_{36},~f_{47}^-f_{28}f_{55}f_{52}^-f_{7}^-,\]

\[f_{8}f_{59}f_{11}f_{39}f_{46},~f_{17}f_{23}^-f_{9}f_{6}^-f_{25},~f_{47}^-f_{35}f_{37}^-f_{48}f_{17}^-,~f_{24}f_{35}f_{22}^-f_{41}^-f_{27}^-,~f_{25}^-f_{40}^-f_{21}f_{41}f_{29},~f_{59}^-f_{11}^-f_{39}^-f_{46}^-f_{8}^-,~f_{24}f_{45}^-f_{55}f_{4}^-f_{53},~f_{19}f_{38}^-f_{33}^-f_{34}f_{37},\]

\[f_{3}^-f_{55}f_{42}^-f_{34}f_{54},~f_{2}^-f_{56}f_{41}^-f_{33}f_{54},~f_{24}f_{59}f_{54}^-f_{12}^-f_{49}^-,~f_{18}f_{47}f_{36}^-f_{38}f_{49}^-,~f_{59}^-f_{37}f_{26}^-f_{52}^-f_{1},~f_{6}^-f_{40}^-f_{43}^-f_{14}^-f_{55}^-,~f_{45}f_{37}f_{11}f_{60}f_{6},~f_{9}^-f_{38}^-f_{41}^-f_{16}^-f_{55}^-,\]

\[f_{51}f_{56}^-f_{27}^-f_{49}f_{6},~f_{51}f_{4}^-f_{57}f_{46}^-f_{25},~f_{6}^-f_{45}^-f_{37}^-f_{11}^-f_{60}^-,~f_{24}f_{30}^-f_{43}^-f_{20}^-f_{37},~f_{23}f_{58}f_{54}^-f_{11}^-f_{48}^-,~f_{10}f_{56}f_{16}f_{42}f_{37},~f_{1}^-f_{59}f_{37}^-f_{26}f_{52},~f_{7}f_{47}f_{28}^-f_{55}^-f_{52},\]

\[f_{51}f_{55}^-f_{26}^-f_{48}f_{5},~f_{23}f_{31}^-f_{43}^-f_{21}^-f_{38},~f_{58}^-f_{54}f_{11}f_{48}f_{23}^-,~f_{23}f_{44}^-f_{56}f_{4}^-f_{52},~f_{10}f_{59}f_{13}f_{36}f_{44},~f_{44}f_{27}f_{30}^-f_{46}^-f_{17}^-,~f_{5}^-f_{51}^-f_{55}f_{26}f_{48}^-,~f_{5}^-f_{39}^-f_{43}^-f_{15}^-f_{56}^-,\]

\[f_{44}f_{56}^-f_{4}f_{52}^-f_{23}^-,~f_{23}f_{36}f_{22}^-f_{42}^-f_{26}^-,~f_{17}f_{29}f_{16}^-f_{14}f_{31}^-,~f_{3}^-f_{12}^-f_{21}^-f_{15}f_{4},~f_{52}f_{14}f_{50}f_{33}^-f_{58}^-,~f_{2}^-f_{11}^-f_{20}^-f_{14}f_{4},~f_{25}^-f_{51}^-f_{4}f_{57}^-f_{46},~f_{9}^-f_{26}f_{20}^-f_{32}^-f_{12},\]

\[f_{10}f_{48}f_{31}^-f_{57}^-f_{53},~f_{49}^-f_{21}f_{50}f_{42}^-f_{45},~f_{7}f_{58}f_{12}f_{40}f_{46},~f_{6}^-f_{55}^-f_{14}^-f_{43}^-f_{40}^-,~f_{17}f_{44}^-f_{27}^-f_{30}f_{46},~f_{59}^-f_{54}f_{12}f_{49}f_{24}^-,~f_{23}f_{38}f_{21}^-f_{43}^-f_{31}^-,~f_{25}^-f_{29}f_{41}f_{21}f_{40}^-,\]

\[f_{8}f_{47}f_{29}^-f_{56}^-f_{53},~f_{59}^-f_{35}f_{28}^-f_{51}^-f_{2},~f_{24}f_{37}f_{20}^-f_{43}^-f_{30}^-,~f_{18}f_{28}f_{16}^-f_{15}f_{30}^-,~f_{8}f_{13}^-f_{33}f_{21}f_{30}^-,~f_{48}^-f_{20}f_{50}f_{41}^-f_{44},~f_{8}f_{35}f_{42}f_{14}f_{57},~f_{7}f_{36}f_{41}f_{15}f_{57},\]

\[f_{60}^-f_{40}f_{30}^-f_{53}^-f_{2},~f_{18}f_{35}f_{34}f_{32}^-f_{40}^-,~f_{19}f_{48}f_{39}^-f_{40}f_{49}^-,~f_{2}^-f_{59}f_{35}^-f_{28}f_{51},~f_{10}f_{37}f_{42}f_{16}f_{56},~f_{51}f_{16}f_{50}f_{32}^-f_{60}^-,~f_{5}^-f_{56}^-f_{15}^-f_{43}^-f_{39}^-,~f_{47}^-f_{22}f_{50}f_{43}^-f_{46},\]

\[f_{48}^-f_{39}f_{40}^-f_{49}f_{19}^-,~f_{48}^-f_{31}f_{57}f_{53}^-f_{10}^-,~f_{47}^-f_{29}f_{56}f_{53}^-f_{8}^-,~f_{19}f_{45}^-f_{28}^-f_{29}f_{44},~f_{1}^-f_{58}f_{38}^-f_{27}f_{53},~f_{19}f_{26}f_{14}^-f_{15}f_{27}^-,~f_{60}^-f_{39}f_{31}^-f_{52}^-f_{3},~f_{60}^-f_{32}^-f_{50}f_{16}f_{51},\]

\[f_{45}f_{55}^-f_{4}f_{53}^-f_{24}^-,~f_{45}f_{26}f_{31}^-f_{46}^-f_{18}^-,~f_{25}^-f_{47}f_{13}f_{54}f_{60}^-,~f_{19}f_{37}f_{34}f_{33}^-f_{38}^-,~f_{58}^-f_{33}^-f_{50}f_{14}f_{52},~f_{46}f_{43}^-f_{50}f_{22}f_{47}^-,~f_{25}^-f_{28}f_{42}f_{20}f_{39}^-,~f_{6}^-f_{51}^-f_{56}f_{27}f_{49}^-,\]

\[f_{58}^-f_{38}f_{27}^-f_{53}^-f_{1},~f_{58}^-f_{12}^-f_{40}^-f_{46}^-f_{7}^-,~f_{1}^-f_{54}f_{32}f_{43}^-f_{57},~f_{17}f_{47}f_{35}^-f_{37}f_{48}^-,~f_{7}f_{13}^-f_{34}f_{20}f_{31}^-,~f_{5}^-f_{29}f_{22}^-f_{34}^-f_{11},~f_{5}^-f_{44}^-f_{38}^-f_{12}^-f_{60}^-,~f_{3}f_{55}^-f_{42}f_{34}^-f_{54}^-\}.\]}

\normalsize{We} verify the following relations among generators:

\tiny{\[f_1=f_{52}f_{14}f_{50}f_{33}^-f_{38}^-f_{27}f_{53},~
f_5=f_{17}^-f_{23}f_{9}^-f_{3}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10},~f_6=f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-,\]

\[f_7=f_{53}^-f_{27}^-f_{38}f_{33}f_{50}^-f_{14}^-f_{52}^-f_{2}f_{17}^-f_{23}f_{9}^-f_{3}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10}f_{17},\]

\[f_8=f_{53}^-f_{27}^-f_{38}f_{33}f_{50}^-f_{14}^-f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-f_{24}f_{10}^-f_{17}^-f_{23}f_{9}^-f_{3}^-f_{52}f_{26}f_{37}^-\]
\[f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10}f_{17}f_{23}^-f_{9}f_{49}f_{27}^-f_{10}f_{37}f_{26}^-f_{23}f_{36}f_{22}^-f_{50}^-f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3},\]

\[f_{11}=f_{2}^-f_{4}f_{14}f_{21}^-f_{38}f_{23}f_{31}^-f_{30}f_{24}^-f_{37}^-,~f_{12}=f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9},~f_{13}=f_{53}^-f_{27}^-f_{38}f_{33}f_{50}^-f_{14}^-f_{52}^-f_{4}f_{22}f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{4}^-f_{22}^-,\]

\[f_{15}=f_{4}^-f_{3}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{21},~f_{16}=f_{22}f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{4}^-,\]

\[f_{18}=f_{24}f_{10}^-f_{17}^-f_{23}f_{9}^-f_{3}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10}f_{17}f_{23}^-f_{9}f_{49}f_{27}^-f_{10}f_{37}f_{26}^-f_{23}f_{36}f_{22}^-f_{50}^-f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3},~f_{19}=f_{10}f_{2}f_{3}^-f_{9}^-,\]

\[f_{20}=f_{37}f_{24}f_{30}^-f_{31}f_{23}^-f_{38}^-f_{21},~f_{25}=f_{17}^-f_{23}f_{9}^-f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-,\]

\[f_{28}=f_{22}f_{33}f_{9}^-f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-,f_{29}=f_{22}f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{4}^-f_{14}^-f_{31}f_{17}^-,\]

\[f_{32}=f_{21}^-f_{38}f_{23}f_{31}^-f_{30}f_{24}^-f_{37}^-f_{26}f_{9}^-f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9},~f_{34}=f_{33}f_{38}f_{9}f_{3}f_{2}^-f_{10}^-f_{37}^-,\]

\[f_{35}=f_{22}f_{44}f_{10}f_{53}f_{30}f_{49}^-f_{9}^-f_{52}^-f_{23}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{37}f_{50}f_{27}f_{24}^-,~f_{39}=f_{9}^-f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{9}f_{26}^-f_{37}f_{24}f_{30}^-f_{31}f_{23}^-f_{38}^-f_{21}f_{33}f_{36}f_{17},\]

\[f_{40}=f_{30}f_{53}f_{2}^-f_{9}^-f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{38}^-f_{44}^-f_{10}^-f_{24}f_{30}^-f_{21}f_{33}f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{3}f_{9}f_{23}^-f_{17},\]

\[f_{41}=f_{44}f_{10}f_{53}f_{30}f_{49}^-f_{9}^-f_{52}^-f_{23}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{37}f_{50},~f_{42}=f_{22}^-f_{36}f_{23}f_{26}^-,~f_{43}=f_{21}^-f_{38}f_{23}f_{31}^-,~f_{45}=f_{22}^-f_{36}f_{23}f_{26}^-f_{50}^-f_{21}^-f_{49},\]

\[f_{46}=f_{31}^-f_{26}f_{22}^-f_{36}f_{23}f_{26}^-f_{50}^-f_{21}^-f_{49}f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-f_{9}^-f_{23}f_{17}^-f_{10}^-f_{24}f_{30}^-f_{21}f_{33}f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{3}f_{9}f_{23}^-f_{17}f_{10}f_{24}^-,\]

\[f_{47}=f_{36}f_{38}^-f_{49}f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-f_{9}^-f_{23}f_{17}^-f_{10}^-f_{24}f_{30}^-f_{21}f_{33}f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{3}f_{9}f_{23}^-f_{17}f_{10}f_{24}^-,\]

\[f_{48}=f_{44}f_{50}^-f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{52}f_{9}f_{49}f_{30}^-f_{53}^-f_{10}^-f_{44}^-f_{50}f_{37}f_{24}f_{30}^-f_{31}f_{23}^-f_{38}^-f_{21},~f_{51}=f_{4}f_{52}^-f_{23}^-f_{44}f_{10}f_{37}f_{26}^-f_{23}f_{36}f_{22}^-f_{50}^-f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3},\]

\[f_{54}=f_{9}^-f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{49}^-f_{24}f_{37}f_{26}^-f_{14}f_{50}f_{33}^-f_{38}^-f_{27}f_{53},~f_{55}=f_{4}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{22}^-f_{50}^-f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{52}f_{9}f_{49}f_{30}^-f_{53}^-f_{10}^-f_{44}^-f_{38}^-f_{9}^-,\]

\[f_{56}=f_{44}f_{10}f_{53}f_{30}f_{49}^-f_{9}^-f_{52}^-f_{23}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{37}f_{50}f_{33}^-f_{53}^-f_{27}^-f_{38}f_{33}f_{50}^-f_{14}^-f_{26}f_{37}^-f_{24}^-f_{49}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{2},~f_{57}=f_{30}^-f_{49}f_{9}f_{52},\]

\[f_{58}=f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{44}^-f_{23}f_{52}f_{4}^-f_{17}f_{31}^-f_{14}f_{4}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{22}^-f_{36},\]

\[f_{59}=f_{53}f_{4}^-f_{3}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{21}f_{50}f_{37}f_{10}f_{2}f_{3}^-f_{9}^-f_{38}^-f_{33}^-,\]

\[f_{60}=f_{9}^-f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{38}^-f_{44}^-f_{10}^-f_{24}f_{30}^-f_{21}f_{33}f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{3}f_{9}f_{23}^-f_{17}.\]}

\normalsize{As for} $H_3$, replacing the generators on the left hand sides by the words on the right hand side yields a presentation $\pi_1(\mathbf{T}(H_4))=\left<\Psi_4'~|~R_4'\right>$ where
\[\Psi_4'=\{f_i,~i\in\underbrace{\{2,3,4,9,10,14,17,21,22,23,24,26,27,30,31,33,36,37,38,44,49,50,52,53\}}_{=:I_{24}}\},\]
and the relations of $R_4'$ become all trivial in the abelianization and thus $\pi_1(\mathbf{T}(H_4))={\mathbb Z}^{24}$.

\iffalse
\tiny{\[R_4':=\{f_{23}f_{31}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{21}^-f_{38},~f_{9}f_{3}f_{2}^-f_{10}^-f_{9}^-f_{3}^-f_{2}f_{10},~f_{22}^-f_{36}f_{23}f_{26}^-f_{22}f_{36}^-f_{23}^-f_{26},~f_{23}f_{44}^-f_{4}f_{52}^-f_{23}^-f_{44}f_{4}^-f_{52},~f_{30}^-f_{49}f_{9}f_{52}f_{30}f_{49}^-f_{9}^-f_{52}^-,~f_{44}f_{27}f_{30}^-f_{17}f_{44}^-f_{27}^-f_{30}f_{17}^-,\]

\[f_{52}^-f_{14}^-f_{50}^-f_{33}f_{52}f_{14}f_{50}f_{33}^-,~f_{27}^-f_{44}^-f_{23}f_{9}^-f_{49}^-f_{27}f_{44}f_{23}^-f_{9}f_{49},~f_{10}f_{37}f_{26}^-f_{23}f_{44}^-f_{10}^-f_{37}^-f_{26}f_{23}^-f_{44},~f_{9}f_{49}f_{24}^-f_{37}^-f_{26}f_{9}^-f_{49}^-f_{24}f_{37}f_{26}^-,~f_{49}f_{38}^-f_{44}^-f_{10}^-f_{24}f_{49}^-f_{38}f_{44}f_{10}f_{24}^-,\]

\[f_{36}^-f_{22}f_{50}f_{21}f_{38}^-f_{36}f_{22}^-f_{50}^-f_{21}^-f_{38},~f_{21}^-f_{33}^-f_{36}^-f_{44}^-f_{27}^-f_{21}f_{33}f_{36}f_{44}f_{27},~f_{3}^-f_{9}^-f_{26}f_{14}^-f_{4}^-f_{3}f_{9}f_{26}^-f_{14}f_{4},~f_{14}f_{50}f_{33}^-f_{36}^-f_{17}^-f_{31}f_{14}^-f_{50}^-f_{33}f_{36}f_{17}f_{31}^-,~f_{53}^-f_{30}^-f_{49}f_{9}f_{3}f_{2}^-f_{53}f_{30}f_{49}^-f_{9}^-f_{3}^-f_{2},\]

\[f_{37}f_{24}f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{37}^-f_{24}^-f_{30}f_{31}^-f_{23}f_{38}f_{21}^-,~f_{49}f_{21}^-f_{50}^-f_{26}^-f_{23}f_{36}f_{22}^-f_{49}^-f_{21}f_{50}f_{26}f_{23}^-f_{36}^-f_{22},~f_{3}^-f_{9}^-f_{27}^-f_{14}f_{26}^-f_{10}^-f_{2}^-f_{3}f_{9}f_{27}f_{14}^-f_{26}f_{10}f_{2},\]

\[f_{52}^-f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{53}^-f_{52}f_{14}f_{50}f_{33}^-f_{38}^-f_{27}f_{53},~f_{10}f_{2}f_{3}^-f_{9}^-f_{37}f_{33}f_{38}f_{9}f_{3}f_{2}^-f_{10}^-f_{37}^-f_{33}^-f_{38}^-,~f_{52}^-f_{9}^-f_{49}^-f_{30}f_{53}f_{10}f_{31}f_{52}f_{9}f_{49}f_{30}^-f_{53}^-f_{10}^-f_{31}^-,\]

\[f_{10}f_{2}f_{3}^-f_{9}^-f_{26}f_{31}^-f_{17}f_{44}^-f_{10}^-f_{2}^-f_{3}f_{9}f_{26}^-f_{31}f_{17}^-f_{44},~f_{53}^-f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{14}^-f_{4}^-f_{53}f_{30}f_{21}^-f_{38}f_{23}f_{31}^-f_{14}f_{4},~f_{3}^-f_{9}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{9}f_{3}f_{2}^-f_{10}^-f_{38}f_{21}^-f_{30}f_{24}^-f_{10}f_{2},\]

\[f_{10}f_{2}f_{4}^-f_{14}^-f_{26}f_{9}^-f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{21}f_{27}^-,~f_{37}^-f_{26}f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{53}^-f_{37}f_{26}^-f_{14}f_{50}f_{33}^-f_{38}^-f_{27}f_{53},~f_{22}^-f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{4}f_{22}f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{4}^-,\]

\[f_{21}f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{21}^-f_{50}^-f_{22}^-f_{36}f_{23}f_{26}^-f_{37}f_{10}f_{27}^-,~f_{38}^-f_{33}^-f_{50}f_{14}f_{26}^-f_{10}^-f_{2}^-f_{3}f_{9}f_{38}f_{33}f_{50}^-f_{14}^-f_{26}f_{10}f_{2}f_{3}^-f_{9}^-,~f_{33}^-f_{36}^-f_{17}^-f_{31}f_{26}^-f_{37}f_{10}f_{27}^-f_{21}f_{33}f_{36}f_{17}f_{31}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{21}^-,\]

\[f_{49}^-f_{9}^-f_{52}^-f_{53}f_{27}f_{38}^-f_{23}^-f_{10}^-f_{53}^-f_{52}f_{9}f_{49}f_{30}^-f_{31}f_{10}f_{27}^-f_{38}f_{23}f_{31}^-f_{30},~f_{2}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{30}^-f_{24}f_{37}f_{14}^-f_{4}^-f_{2}f_{31}^-f_{23}f_{38}f_{21}^-f_{30}f_{24}^-f_{37}^-f_{14}f_{4},\]

\[f_{53}f_{4}^-f_{22}^-f_{36}f_{23}f_{26}^-f_{50}^-f_{21}^-f_{49}f_{24}^-f_{53}^-f_{4}f_{22}f_{36}^-f_{23}^-f_{26}f_{50}f_{21}f_{49}^-f_{24},~f_{23}^-f_{36}^-f_{33}^-f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{23}f_{36}f_{33}f_{21}f_{27}^-f_{10}f_{2}f_{4}^-f_{14}^-f_{26},\]

\[f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{37}f_{10}f_{2}f_{3}^-f_{4}f_{53}^-f_{24}^-f_{49}f_{21}^-f_{50}^-f_{37}^-f_{10}^-f_{2}^-f_{3},~f_{36}^-f_{38}f_{49}^-f_{30}f_{53}f_{2}^-f_{3}f_{52}^-f_{31}^-f_{17}f_{36}f_{38}^-f_{49}f_{30}^-f_{53}^-f_{2}f_{3}^-f_{52}f_{31}f_{17}^-,\]

\[f_{53}^-f_{30}^-f_{49}f_{9}f_{52}f_{31}f_{44}^-f_{38}^-f_{9}^-f_{26}f_{31}^-f_{52}^-f_{9}^-f_{49}^-f_{30}f_{53}f_{10}f_{44}f_{38}f_{9}f_{26}^-f_{10}^-,~f_{17}f_{4}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{22}f_{14}^-f_{31}f_{17}^-f_{4}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{22}^-f_{14}f_{31}^-,\]

\[f_{27}^-f_{10}f_{2}f_{4}^-f_{14}^-f_{37}f_{24}f_{30}^-f_{31}f_{23}^-f_{38}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{37}^-f_{24}^-f_{30}f_{31}^-f_{23}f_{38},~f_{23}^-f_{52}^-f_{3}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{38}f_{23}f_{52}f_{3}^-f_{21}f_{27}^-f_{10}f_{2}f_{4}^-f_{14}^-f_{26}f_{9}^-f_{38}^-,\]

\[f_{9}f_{38}f_{33}f_{2}f_{10}f_{37}f_{50}f_{21}f_{49}^-f_{24}f_{53}f_{4}^-f_{9}^-f_{38}^-f_{33}^-f_{2}^-f_{10}^-f_{37}^-f_{50}^-f_{21}^-f_{49}f_{24}^-f_{53}^-f_{4},~f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10}f_{2}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-f_{24}f_{10}^-f_{2}^-,\]

\[f_{50}^-f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{30}^-f_{49}f_{9}f_{52}f_{53}^-f_{10}^-f_{23}^-f_{52}^-f_{4}f_{10}f_{37}f_{50}f_{21}f_{49}^-f_{24}f_{53}f_{4}^-f_{9}^-f_{38}^-,~f_{4}f_{52}^-f_{23}^-f_{10}^-f_{53}^-f_{52}f_{9}f_{49}f_{30}^-f_{23}f_{38}f_{21}^-f_{30}f_{24}^-f_{37}^-f_{50}^-f_{38}^-f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{37}f_{10},\]

\[f_{17}^-f_{31}f_{14}^-f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{38}^-f_{36}f_{17}f_{31}^-f_{14}f_{50}f_{21}f_{27}^-f_{10}f_{37}f_{26}^-f_{23}f_{38}f_{21}^-f_{50}^-f_{22}^-,~f_{14}f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{52}f_{9}f_{49}f_{30}^-f_{53}^-f_{10}^-f_{23}^-f_{52}^-f_{14}^-f_{50}^-f_{38}^-f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{37}f_{10}f_{4},\]

\[f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{14}^-f_{31}f_{23}^-f_{9}f_{49}f_{21}^-f_{50}^-f_{22}^-f_{36}f_{23}f_{26}^-f_{24}^-f_{53}^-f_{4}f_{14}f_{31}^-f_{23},~f_{38}f_{44}f_{23}^-f_{52}^-f_{4}f_{10}f_{37}f_{50}f_{21}f_{49}^-f_{24}f_{53}f_{4}^-f_{9}^-f_{38}^-f_{44}^-f_{23}f_{52}f_{4}^-f_{10}^-f_{37}^-f_{50}^-f_{21}^-f_{49}f_{24}^-f_{53}^-f_{4}f_{9},\]

\[f_{9}^-f_{49}^-f_{21}f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{53}^-f_{30}^-f_{49}f_{9}f_{52}f_{14}f_{22}^-f_{36}f_{23}f_{26}^-f_{37}f_{10}f_{53}f_{30}f_{21}^-f_{50}^-f_{14}^-f_{52}^-,\]

\[f_{30}f_{31}^-f_{23}f_{36}f_{22}^-f_{50}^-f_{21}^-f_{38}f_{44}f_{10}f_{37}f_{26}^-f_{9}f_{49}f_{30}^-f_{31}f_{23}^-f_{38}^-f_{21}f_{50}f_{22}f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{9}^-f_{49}^-,\]

\[f_{37}f_{10}f_{2}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{37}^-f_{24}^-f_{49}f_{9}f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{53}^-f_{33}^-f_{38}^-f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50},\]

\[f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{50}f_{22}f_{30}f_{21}^-f_{33}^-f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{10}f_{37}f_{22}^-f_{36}f_{23}f_{26}^-f_{14}f_{31}^-f_{44}f_{27},\]

\[f_{4}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{22}f_{50}f_{33}^-f_{36}^-f_{17}^-f_{31}f_{52}f_{3}^-f_{22}^-f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{4}f_{50}^-f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3},\]

\[f_{52}^-f_{31}^-f_{17}f_{36}f_{33}f_{50}^-f_{22}^-f_{36}f_{23}f_{26}^-f_{37}f_{10}f_{27}^-f_{49}f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-f_{3},\]

\[f_{3}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-f_{24}f_{10}^-f_{17}^-f_{23}f_{9}^-f_{3}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10}f_{17}f_{23}^-f_{9},\]

\[f_{3}f_{52}^-f_{31}^-f_{17}f_{36}f_{33}f_{50}^-f_{4}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{22}^-f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{50}f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{4}^-f_{52}f_{23}f_{44}^-,\]

\[f_{21}f_{17}^-f_{36}^-f_{33}^-f_{2}^-f_{4}f_{14}f_{37}^-f_{24}^-f_{30}f_{31}^-f_{23}f_{38}f_{21}^-f_{10}^-f_{27}f_{21}^-f_{17}f_{36}f_{33}f_{2}f_{4}^-f_{14}^-f_{37}f_{24}f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{10}f_{27}^-,\]

\[f_{17}f_{23}^-f_{9}f_{3}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-f_{24}f_{10}^-f_{44}^-f_{38}^-f_{9}^-f_{26}f_{37}^-f_{24}^-f_{30}f_{31}^-f_{23}f_{38}f_{21}^-f_{33}^-f_{36}^-f_{17}^-f_{31}f_{52}f_{3}^-,\]

\[f_{10}f_{24}^-f_{30}f_{21}^-f_{33}^-f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{3}^-f_{9}^-f_{23}f_{17}^-f_{44}f_{38}f_{9}f_{26}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{30}^-f_{24}f_{37}f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3},\]

\[f_{14}f_{4}f_{2}^-f_{10}^-f_{27}f_{21}^-f_{17}f_{36}f_{33}f_{31}f_{23}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{37}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{17}^-f_{36}^-f_{33}^-f_{37}^-f_{24}^-f_{30}f_{31}^-f_{23}f_{38}f_{21}^-,\]

\[f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{44}f_{36}f_{33}f_{21}f_{27}^-f_{10}f_{37}f_{26}^-f_{52}^-f_{3}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{33}^-f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{3}^-f_{9}^-,\]

\[f_{24}f_{22}f_{38}^-f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{37}f_{10}f_{44}f_{23}^-f_{52}^-f_{4}f_{27}f_{24}^-f_{22}^-f_{50}^-f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{30}^-f_{49}f_{9}f_{52}f_{53}^-f_{10}^-f_{44}^-f_{27}^-,\]

\[f_{44}f_{10}f_{2}f_{3}^-f_{9}^-f_{49}^-f_{21}f_{50}f_{37}f_{10}f_{44}f_{23}^-f_{52}^-f_{4}f_{14}f_{31}^-f_{17}f_{44}^-f_{9}f_{3}f_{2}^-f_{10}^-f_{49}f_{21}^-f_{50}^-f_{37}^-f_{10}^-f_{44}^-f_{23}f_{52}f_{4}^-f_{14}^-f_{31}f_{17}^-,\]

\[f_{52}^-f_{23}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{37}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{33}f_{36}f_{17}f_{3}^-f_{52}f_{23}f_{38}f_{21}^-f_{30}f_{24}^-f_{37}^-f_{14}f_{4}f_{2}^-f_{10}^-f_{27}f_{21}^-f_{33}^-f_{36}^-f_{17}^-f_{3},\]

\[f_{37}^-f_{31}f_{30}^-f_{49}f_{9}f_{52}f_{53}^-f_{10}^-f_{17}^-f_{22}^-f_{50}^-f_{21}^-f_{38}f_{23}f_{31}^-f_{17}f_{44}^-f_{27}^-f_{30}f_{22}f_{4}f_{52}^-f_{23}^-f_{44}f_{10}f_{37}f_{50}f_{21}f_{49}^-f_{24}f_{53}f_{4}^-f_{9}^-f_{38}^-f_{27}f_{24}^-,\]

\[f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-f_{24}f_{9}f_{3}f_{2}^-f_{10}^-f_{23}^-f_{31}f_{14}^-f_{4}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{9}^-f_{3}^-f_{2}f_{10}f_{23}f_{31}^-f_{14}f_{4}f_{52}^-,\]

\[f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{14}f_{50}f_{33}^-f_{38}^-f_{27}f_{53}f_{36}^-f_{44}^-f_{10}^-f_{24}f_{49}^-f_{21}f_{14}^-f_{26}f_{9}^-f_{3}^-f_{2}f_{10}f_{27}^-f_{53}^-f_{24}^-f_{49}f_{21}^-f_{50}^-f_{33}f_{38}f_{10}^-f_{2}^-f_{3}f_{9}f_{37}^-,\]

\[f_{4}^-f_{52}f_{23}f_{44}^-f_{27}^-f_{10}f_{2}f_{3}^-f_{9}^-f_{26}f_{14}^-f_{30}f_{24}^-f_{37}^-f_{14}f_{4}f_{2}^-f_{10}^-f_{27}f_{21}^-f_{33}^-f_{36}^-f_{23}^-f_{9}f_{3}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-f_{24}f_{10}^-,\]

\[f_{50}f_{21}f_{38}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-f_{9}^-f_{23}f_{17}^-f_{49}f_{21}^-f_{50}^-f_{22}^-f_{36}f_{23}f_{31}^-f_{17}f_{44}^-f_{27}^-f_{30}f_{24}^-f_{10}f_{44}f_{38}f_{9}f_{26}^-f_{37}f_{24}f_{30}^-f_{31}f_{23}^-f_{36}^-f_{22},\]

\[f_{4}^-f_{14}^-f_{31}f_{23}^-f_{38}^-f_{21}f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{33}^-f_{21}^-f_{27}f_{10}^-f_{37}^-f_{24}^-f_{53}^-f_{4}f_{14}f_{31}^-f_{23}f_{38}f_{21}^-f_{3}f_{52}^-f_{31}^-f_{17}f_{36}f_{33}f_{21}f_{27}^-f_{10}f_{37}f_{24}f_{53},\]

\[f_{17}^-f_{44}f_{27}f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{50}f_{22}f_{49}f_{38}^-f_{36}f_{17}f_{31}^-f_{52}^-f_{9}^-f_{49}^-f_{30}f_{53}f_{10}f_{37}f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{53}^-f_{27}^-f_{38}f_{9}f_{26}^-f_{23}f_{38}f_{21}^-f_{50}^-f_{22}^-,\]

\newpage

\[f_{38}^-f_{9}^-f_{26}f_{37}^-f_{24}^-f_{30}f_{31}^-f_{23}f_{38}f_{21}^-f_{33}^-f_{36}^-f_{17}^-f_{31}f_{52}f_{3}^-f_{2}f_{53}^-f_{27}^-f_{44}^-f_{17}f_{23}^-f_{9}f_{3}f_{52}^-f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-f_{24}f_{10}^-f_{2}^-f_{53}f_{27},\]

\[f_{53}^-f_{27}^-f_{38}f_{36}^-f_{17}^-f_{31}f_{14}^-f_{22}f_{36}^-f_{23}^-f_{26}f_{37}^-f_{10}^-f_{27}f_{49}^-f_{30}f_{27}^-f_{44}^-f_{17}f_{31}^-f_{23}f_{36}f_{22}^-f_{50}^-f_{21}^-f_{49}f_{53}f_{27}f_{38}^-f_{33}^-f_{50}f_{14}f_{26}^-f_{37}f_{10}f_{44}f_{36}f_{33}f_{21}f_{30}^-,\]

\[f_{27}^-f_{38}f_{9}f_{4}f_{53}^-f_{24}^-f_{49}f_{21}^-f_{50}^-f_{37}^-f_{10}^-f_{4}^-f_{52}f_{23}f_{44}^-f_{33}f_{9}^-f_{26}f_{14}^-f_{4}^-f_{2}f_{10}f_{27}^-f_{21}f_{49}^-f_{27}f_{44}f_{23}^-f_{52}^-f_{4}f_{2}^-f_{53}f_{27}f_{38}^-f_{33}^-f_{50}f_{14}f_{26}^-f_{37}f_{24},\]

\[f_{31}f_{14}^-f_{4}^-f_{53}f_{24}f_{22}f_{36}^-f_{23}^-f_{26}f_{50}f_{21}f_{49}^-f_{9}^-f_{38}^-f_{21}f_{30}^-f_{53}^-f_{2}f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{26}^-f_{9}f_{49}f_{24}^-f_{10}f_{44}f_{36}f_{22}^-f_{50}^-f_{31}^-f_{23}f_{38}f_{21}^-f_{30}f_{27}^-f_{44}^-,\]

\[f_{23}f_{9}^-f_{3}^-f_{2}f_{10}f_{24}^-f_{30}f_{21}^-f_{33}^-f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{53}^-f_{23}^-f_{9}f_{3}f_{2}^-f_{10}^-f_{24}f_{30}^-f_{21}f_{33}f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{14}f_{50}f_{33}^-f_{38}^-f_{27}f_{53},\]

\[f_{27}f_{38}^-f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{37}f_{10}f_{44}f_{23}^-f_{52}^-f_{4}f_{36}f_{23}f_{10}f_{2}f_{3}^-f_{9}^-f_{27}^-f_{50}^-f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{30}^-f_{49}f_{9}f_{52}f_{53}^-f_{10}^-f_{44}^-f_{36}^-f_{23}^-f_{9}f_{3}f_{2}^-f_{10}^-,\]

\[f_{53}^-f_{2}f_{3}^-f_{52}f_{31}f_{17}^-f_{36}^-f_{38}f_{10}^-f_{2}^-f_{3}f_{9}f_{37}^-f_{36}f_{44}f_{10}f_{37}f_{26}^-f_{52}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{26}f_{23}^-f_{36}^-f_{22}f_{9}^-f_{38}^-f_{27}f_{24}^-f_{49}f_{38}^-f_{36}f_{22}^-f_{50}^-f_{31}^-f_{23}f_{38}f_{21}^-f_{17}f_{44}^-f_{27}^-,\]

\[f_{36}^-f_{44}^-f_{10}^-f_{37}^-f_{26}f_{52}f_{4}^-f_{14}^-f_{31}f_{17}^-f_{44}f_{10}f_{2}f_{3}^-f_{9}^-f_{24}^-f_{53}^-f_{4}f_{14}f_{50}f_{33}^-f_{9}f_{26}^-f_{37}f_{24}f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3}f_{2}^-f_{53}f_{30}f_{31}^-f_{23}f_{38}f_{21}^-f_{50}^-,\]

\[f_{37}^-f_{26}f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{53}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{9}^-f_{3}^-f_{2}f_{10}f_{23}f_{52}f_{9}f_{49}f_{30}^-f_{27}f_{10}^-f_{2}^-f_{3}f_{9}f_{26}^-f_{14}f_{50}f_{37}f_{24}f_{30}^-f_{21}f_{38}^-f_{23}^-f_{52}^-f_{9}^-f_{49}^-f_{30}f_{53}f_{10}f_{44}f_{36},\]

\[f_{36}f_{38}^-f_{9}^-f_{4}^-f_{53}f_{24}f_{49}^-f_{21}f_{50}f_{37}f_{10}f_{44}f_{23}^-f_{52}^-f_{4}f_{27}f_{10}^-f_{2}^-f_{3}f_{9}f_{26}^-f_{14}f_{30}^-f_{49}f_{9}f_{14}^-f_{50}^-f_{33}f_{38}f_{27}^-f_{53}^-f_{52}f_{26}f_{37}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{21}^-f_{30}f_{24}^-f_{10}f_{2}f_{3}^-f_{9}^-f_{23},\]

\[f_{14}^-f_{26}f_{10}f_{2}f_{3}^-f_{9}^-f_{27}^-f_{50}^-f_{37}^-f_{24}^-f_{30}f_{21}^-f_{38}f_{23}f_{30}^-f_{49}f_{9}f_{52}f_{53}^-f_{10}^-f_{44}^-f_{36}^-f_{33}^-f_{50}f_{14}f_{52}f_{9}f_{26}^-f_{31}f_{23}^-f_{38}^-f_{21}f_{30}^-f_{24}f_{37}f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3}f_{2}^-f_{53}f_{30}f_{17}^-f_{44}f_{27}f_{49}^-f_{9}^-f_{52}^-,\]

\[f_{49}f_{27}^-f_{10}f_{37}f_{22}^-f_{36}f_{23}f_{26}^-f_{50}^-f_{33}f_{36}f_{17}f_{31}^-f_{52}^-f_{3}f_{9}f_{23}^-f_{36}^-f_{33}^-f_{21}^-f_{27}f_{10}^-f_{2}^-f_{4}f_{14}f_{50}f_{21}f_{49}^-f_{24}f_{53}f_{4}^-f_{9}^-f_{49}^-\]
\[f_{30}f_{14}^-f_{26}f_{10}f_{2}f_{3}^-f_{9}^-f_{27}^-f_{38}f_{9}f_{4}f_{53}^-f_{24}^-f_{49}f_{21}^-f_{50}^-f_{37}^-f_{10}^-f_{44}^-f_{23}f_{52}f_{4}^-f_{36}^-f_{17}^-f_{44}f_{27}f_{30}^-f_{21}f_{38}^-f_{23}^-f_{31}f_{50}f_{22}\}.\]}
\fi

\normalsize{In} terms of $H_4$-conjugates of $q_0:=\widehat{s}_0r_{H_4}$, the 24 generators $\{f_i,~i\in I_{24}\}$ of $\pi_1(\mathbf{T}(H_4))$ given above may be written $f_i=q_0^{u_i}$ where the $u_i$'s are given by

\[\begin{array}{c}
\{u_i,~i\in I_{24}\}=\{(xys_4)^4,~(s_4xy)^4s_4=r_{H_4},~(s_1(ys_4s_3)^{s_2})^3s_1s_2ys_4,~xys_ 4,~(xy)^{s_4},~t^{s_2}ys_4,~ts_2y(xy)^{s_4}, \\[.5em]

(s_2(s_4x)^{s_3}s_2^{s_1})^2s_2s_3s_4,~s_2zy(xy)^{s_4},~xzys_4,~(xzy)^{s_4},~xts_2ys_4,~(xts_2y)^{s_4},~(s_4xs_3)^2s_3ys_4, \\[.5em]

xs_3(xy)^{s_4},~s_1t^{s_2}ys_4,~(s_4^xs_3s_1)^2x^{-1}s_4,~xts_2y(xy)^{s_4},~s_4xts_2y(xy)^{s_4},xzy(xy)^{s_4}, \\[.5em]

(xzy)^{s_4}s_1^{s_2}ys_4,~s_1s_2zys_4s_1^{s_2}ys_4,~((ys_4x)^{s_3})^3s_3ys_4,~x_3ys_4(s_1^{s_2}ys_4s_3)^2s_1^{s_2}ys_4\},\end{array}\]
with
\[x:=s_3s_1^{s_2},~y:=((s_1s_2)^2)^{s_3},~z:=s_3s_4s_1s_2,~t:=s_3s_4s_2s_1.\]
\newpage


\section{Hyperbolic extensions of finite irreducible Weyl groups}

As the irreducible hyperbolic Coxeter groups have rank $\le10$ and are all classified (see \cite{chein}), we can check each reflection of each irreducible finite Weyl group to see which one of them give hyperbolic extensions. There may are other possible reflections and extensions, but the resulting Coxeter diagram must appear in the following table. The computations were made using \cite{GAP4}. Of course, for the case of $G_2=I_2(6)$, we find in particular the diagram corresponding to $\widehat{I_2(6)}$ defined above.
\begin{table}[h!]
\resizebox{1.05\textwidth}{!}{
\begin{tabular}{|c|c|c|c|c|}
\hline
Type & Dynkin diagram & Hyperbolic diagram & Reflection & Compact? \\
\hline
\hline
 & & & & \\
 $G_2$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	
	\draw (a)--(b);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (1/2,0) node[above]{$6$};
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} &
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (-1/2,1);
	
	\draw (-1/2,0) node[below]{$6$};
	\draw (-1/4,1/2) node[right]{$6$};
	
	\draw (a)--(b);
	\draw (b)--(c);
	\draw (a)--(c);
	
	\fill[fill=white] (c) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture} \\ 
and \\
\begin{tikzpicture}[rotate=180]
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (-2,0);
	
	\draw (-1/2,0) node[above]{$6$};

	
	\draw (a)--(b);
	\draw (a)--(c);
	
	\fill[fill=white] (c) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $\begin{array}{c}s_1^{s_2} \\
 \\
\text{and} \\
 \\
s_1^{s_2s_1}\end{array}$ & both \\
 & & & & \\
\hline
 & & & & \\
$A_3$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (-1,0);
	
	\draw (a) node[below]{$2$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$1$};
	
	\draw (a)--(b);
	\draw (a)--(c);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
\end{tikzpicture}\end{minipage} & 
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (0,1);
	\coordinate (d) at (1,1);
	
	\draw (a)--(b);
	\draw (a)--(c);
	\draw (a)--(d);
	\draw (b)--(d);
	\draw (c)--(d);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=white] (d) circle (2.5pt);
	\node[mark size=2.5pt] at (d) {\pgfuseplotmark{otimes}};	
\end{tikzpicture}\end{minipage} & $s_3^{s_2}$ & no \\
 & & & & \\
\hline
 & & & & \\
$C_3$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (-1,0);
	
	\draw (a) node[below]{$2$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$1$};
	
	\draw (a)--(b);
	\draw (a)--(c);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	
	\draw (-1/2,0) node[above]{$4$};
\end{tikzpicture}\end{minipage} & 
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (0,1);
	\coordinate (d) at (1,1);
	
	\draw (a)--(b);
	\draw (a)--(c);
	\draw (b)--(d);
	\draw (c)--(d);
	
	\draw (1/2,1) node[above]{$4$};
	\draw (0,1/2) node[left]{$4$};
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\node[mark size=2.5pt] at (a) {\pgfuseplotmark{otimes}};	
\end{tikzpicture}\end{minipage} & $s_2^{s_1}$ & no \\
 & & & & \\
\hline
 & & & & \\
$C_4$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (-1,0);
	\coordinate (d) at (2,0);
	
	\draw (a) node[below]{$2$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$1$};
	\draw (d) node[below]{$4$};
	
	\draw (-1/2,0) node[above]{$4$};
	
	\draw (a)--(b);
	\draw (a)--(c);
	\draw (c)--(d);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
\end{tikzpicture}\end{minipage} & 
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (0,1);
	\coordinate (d) at (1,1);
	\coordinate (e) at (-1,1);
	
	\draw (a)--(b);
	\draw (a)--(c);
	\draw (c)--(e);
	\draw (c)--(d);
	\draw (b)--(d);
	
	\draw (-1/2,1) node[above]{$4$};
	
	\fill[fill=white] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);	
	\fill[fill=black] (e) circle (2.5pt);
	\node[mark size=2.5pt] at (a) {\pgfuseplotmark{otimes}};
\end{tikzpicture}\end{minipage} & $s_3^{s_1^{s_2}}=s_3^{s_2s_1s_2}$ & no \\
 & & & & \\
\hline
 & & & & \\
$D_4$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-1/2,1.732/2);
	\coordinate (b) at (-1/2,-1.732/2);
	\coordinate (c) at (1/2,0);
	\coordinate (d) at (1+1/2,0);
	\coordinate (e) at (2+1/3,0);
	\coordinate (f) at (3+1/2,0);
	\coordinate (g) at (4+1/2,1.732/2);
	\coordinate (h) at (4+1/2,-1.732/2);
	
	\draw (e) node[below]{$1$};
	\draw (f) node[below]{$2$};
	\draw (g) node[right]{$4$};
	\draw (h) node[right]{$3$};
	
	\draw (e)--(f)
	(f)--(g)
	(f)--(h);
	
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=black] (g) circle (2.5pt);
	\fill[fill=black] (h) circle (2.5pt);
	\draw (g) circle (2.5pt);
	\draw (h) circle (2.5pt);
\end{tikzpicture}\end{minipage} & 
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (0,0);
	\coordinate (b) at (1,0);
	\coordinate (c) at (0,1);
	\coordinate (d) at (1,1);
	\coordinate (e) at (1/2,1/2);
	
	\draw (d) node[right]{$3$};
	\draw (b) node[right]{$2$};
	\draw (a) node[left]{$1$};
	
	\draw (a)--(c)
	(b)--(c)
	(a)--(b)
	(c)--(d)
	(d)--(b);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=white] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
	\node[mark size=2.5pt] at (c) {\pgfuseplotmark{otimes}};
\end{tikzpicture}\end{minipage} & $s_4^{s_2s_3}$ & no \\
 & & & & \\
\hline
 & & & & \\
$F_4$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}[scale=1]
	\coordinate (a) at (-1,0);
	\coordinate (b) at (0,0);
	\coordinate (c) at (1,0);
	\coordinate (d) at (2,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$2$};
	\draw (c) node[below]{$3$};
	\draw (d) node[below]{$4$};
	
	\draw (1/2,0) node[above]{$4$};
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
\end{tikzpicture}\end{minipage} & 
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (1,0);
	\coordinate (b) at (0.309,0.9511);
	\coordinate (c) at (-0.809,0.5878);
	\coordinate (d) at (-0.809,-0.5878);
	\coordinate (e) at (0.309,-0.9511);
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d)
	(d)--(e)
	(e)--(a);
	
	\draw (-1.3,0) node[right]{$4$};
	\draw (0.65,0.65) node[right]{$4$};
	
	\fill[fill=white] (a) circle (2.5pt);
	\node[mark size=2.5pt] at (a) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
\end{tikzpicture}\end{minipage} & $s_3^{s_2s_1}$ & no \\
 & & & & \\
\hline
 & & & & \\
$E_8$ & \begin{minipage}{0.45\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (-2,0);
	\coordinate (b) at (-1,0);
	\coordinate (c) at (0,0);
	\coordinate (d) at (1,0);
	\coordinate (e) at (2,0);
	\coordinate (f) at (0,1);
	\coordinate (g) at (3,0);
	\coordinate (h) at (-3,0);
	\coordinate (i) at (4,0);
	
	\draw (a) node[below]{$1$};
	\draw (b) node[below]{$3$};
	\draw (c) node[below]{$4$};
	\draw (d) node[below]{$5$};
	\draw (e) node[below]{$6$};
	\draw (f) node[above]{$2$};
	\draw (g) node[below]{$7$};
	\draw (i) node[below]{$8$};
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d)
	(d)--(e)
	(c)--(f)
	(e)--(g)
	(g)--(i);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=black] (g) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=black] (d) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=black] (i) circle (2.5pt);
\end{tikzpicture}\end{minipage} & 
\begin{minipage}{0.20\textwidth}\centering \begin{tikzpicture}
	\coordinate (a) at (1,0);
	\coordinate (b) at (0.707,0.707);
	\coordinate (c) at (0,1);
	\coordinate (d) at (-0.707,0.707);
	\coordinate (e) at (-1,0);
	\coordinate (f) at (-0.707,-0.707);
	\coordinate (g) at (0,-1);
	\coordinate (h) at (0.707,-0.707);
	\coordinate (i) at (1.7,0);
	
	\draw (a)--(b)
	(b)--(c)
	(c)--(d)
	(d)--(e)
	(e)--(f)
	(f)--(g)
	(g)--(h)
	(h)--(a)
	(a)--(i);
	
	\fill[fill=black] (a) circle (2.5pt);
	\fill[fill=black] (b) circle (2.5pt);
	\fill[fill=black] (c) circle (2.5pt);
	\fill[fill=white] (d) circle (2.5pt);
	\node[mark size=2.5pt] at (d) {\pgfuseplotmark{otimes}};
	\fill[fill=black] (e) circle (2.5pt);
	\fill[fill=black] (f) circle (2.5pt);
	\fill[fill=black] (g) circle (2.5pt);
	\fill[fill=black] (h) circle (2.5pt);
	\fill[fill=black] (i) circle (2.5pt);
\end{tikzpicture}\end{minipage} &
$s_7^{(s_3s_2)^{s_4s_5s_6}(s_5s_2)^{s_4s_3s_1}}$ & no \\
 & & & & \\
\hline
\end{tabular}}
\captionof{table}{Hyperbolic extensions of finite irreducible Weyl groups.}
\label{otherhyperbolicextensions}
\end{table}
\end{appendix}










\newcommand{\etalchar}[1]{$^{#1}$}
