\section{Introduction}

Since the introduction of the Bost--Connes system in the mid '90s, \cite{BC}, a very rich interplay
between number theory and quantum statistical mechanics developed, involving the Galois theory
of abelian and non-abelian extensions of number fields 
(\cite{CoMaRa}, \cite{Corn1}, \cite{HaPa}, \cite{LLN}, \cite{ManMar3}, \cite{Yalk}),
Shimura varieties (\cite{CoMa}, \cite{HaPa}, \cite{LLN2}), $L$-series and zeta functions 
(\cite{CCM}, \cite{Corn2}), etc. On the other hand, the work of Manin and Marcolli, \cite{ManMar},
and the subsequent work \cite{KesStra}, \cite{ManMar2}, \cite{Mar}, developed a theory of
limiting modular symbols, and a related noncommutative geometry model of the boundary of
modular curves.  It was remarked in \S 7.9 of \cite{CoMa-book} that the noncommutative
compactification of modular curves of \cite{ManMar} should fit as a ``boundary stratum" of
the quantum statistical mechanical system of \cite{CoMa} that generalized the Bost--Connes
system from the case of ${\rm GL}_1$ to the case of ${\rm GL}_2$ and has the geometry of modular curves
directly built into its construction through the Shimura variety of ${\rm GL}_2$. However, the relation 
between the limiting modular symbols of \cite{ManMar} and the 
${\rm GL}_2$ quantum statistical mechanical system of \cite{CoMa} has not been fully 
analyzed. It is the purpose of this note to outline a construction of a ``boundary algebra"
for the ${\rm GL}_2$ system, endowed with an induced time evolution, that is built on the
noncommutative boundary of modular curves described in \cite{ManMar} and on 
limiting modular symbols.

\smallskip

\section{The ${\rm GL}_2$-system}\label{GL2Sec}

In this section we recall the basic properties of the ${\rm GL}_2$-quantum statistical mechanical 
system of \cite{CoMa}, in a version that accounts for the choice of a finite index subgroup
of ${\rm GL}_2({\mathbb Z})$.

\subsection{The ${\rm GL}_2$-quantum statistical mechanical system}

The ${\rm GL}_2$-quantum statistical mechanical system constructed in \cite{CoMa} as a generalization
of the Bost--Conne system of \cite{BC} has algebra of observables given by the non-commutative
$C^*$-algebra describing the ``bad quotient" of the space of $2$-dimensional ${\mathbb Q}$-lattices up
to scaling by the equivalence relation of commensurability. This algebra is made dynamical by a
time evolution defined by the determinant of the ${\rm GL}_2^+({\mathbb Q})$ matrix that implements 
commensurability. There is an arithmetic algebra of unbounded multipliers on the $C^*$-algebra
of observable, which is built in a natural way out of modular functions and Hecke operators
(see \cite{CoMa} and Chapter~3 of \cite{CoMa-book}. The KMS-states for the time evolution
have an action of ${\mathbb Q}^*\backslash {\rm GL}_2({\mathbb A}_{{\mathbb Q},f})$ by symmetries, which include both
automorphisms and endomorphisms of the $C^*$-dynamical system. The KMS states at
zero temperature, defined as weak limits of KMS-states at positive temperature, are evaluations
of modular functions at points in the upper half plane and the induced action of symmetries
on KMS-states recovers the Galois action of the automorphisms of the modular field. 
In the case of imaginary quadratic fields, the associated Bost--Connes system, constructed
in \cite{HaPa} and \cite{LLN}, can be seen as a specialization of the 
${\rm GL}_2$-quantum statistical mechanical system of \cite{CoMa} at $2$-dimensional ${\mathbb Q}$-lattices that
are $1$-dimensional ${\mathbb K}$-lattices, with ${\mathbb K}$ the imaginary quadratic field, and to CM points in the
upper half plane. 

\smallskip

Our goal here is to adapt this construction to obtain a specialization of the ${\rm GL}_2$ system
to the boundary $\P^1({\mathbb R})$ and a further specialization for real quadratic fields.  Our starting
point will be the same algebra of the ${\rm GL}_2$ system of \cite{CoMa}, hence we start by
reviewing briefly that construction in order to use it in our setting.
It is convenient, for the setting we consider below, to extend the construction 
of the ${\rm GL}_2$ system recalled above to the case where we replace ${\rm GL}_2^+({\mathbb Q})$
acting on the upper half plane $\H$ with ${\rm GL}_2({\mathbb Q})$ acting on $\H^\pm$ (the upper and
lower half planes) and we consider a fixed finite index subgroup $G$ of ${\rm GL}_2({\mathbb Z})$,
where the latter replaces $\Gamma={\rm SL}_2({\mathbb Z}) ={\rm GL}_2^+({\mathbb Q})\cap {\rm GL}_2(\hat{\mathbb Z})$ in
the construction of the ${\rm GL}_2$-system. We can formulate the resulting quantum
statistical mechanical system in the following way. 
We can consider two slightly different versions of the convolution algebra. 


\smallskip

\begin{defn}\label{AcG}
The involutive algebra ${\mathcal A}^c_G$ is given by complex valued functions on
\begin{equation}\label{Upm}
 {\mathcal U}^\pm =\{ (g,\rho,z) \in {\rm GL}_2({\mathbb Q}) \times M_2(\hat Z) \times \H^\pm\,|\, g\rho \in M_2(\hat {\mathbb Z}) \} 
\end{equation} 
that are invariant under the action of $G\times G$ by 
$(g,\rho,z) \mapsto (\gamma_1 g \gamma_2^{-1}, \gamma_2 z)$, and that 
have finite support in $g\in G\backslash {\rm GL}_2({\mathbb Q})$, 
depend on the variable $\rho \in M_2(\hat {\mathbb Z})$ through the projection onto some finite
level $p_N: M_2(\hat {\mathbb Z}) \to M_2({\mathbb Z}/N{\mathbb Z})$ and have compact support in the variable $z\in \H^\pm$.
The convolution product on ${\mathcal A}^c_G$ is given by 
\begin{equation}\label{convolG}
(f_1\star f_2)(g,\rho,z) =\sum_{h \in G\backslash {\rm GL}_2({\mathbb Q})\,:\, h\rho \in M_2(\hat{\mathbb Z})} f_1(g h^{-1}, h \rho, h(z)) g_2(h,\rho,z) 
\end{equation}
and the involution is $f^*(g,\rho,z)=\overline{f(g^{-1}, g\rho, g(z))}$.  The algebra ${\mathcal A}^c_G$ is
endowed with a time evolution given by $\sigma_t(f)(g,\rho,z) = |\det(g)|^{it} f(g,\rho,z)$.
\end{defn}

\smallskip

\smallskip

Let $\Xi$ denote the coset space $\Xi={\rm GL}_2({\mathbb Z})\backslash {\rm GL}_2({\mathbb Q}) /{\rm GL}_2({\mathbb Z})$ and let
${\mathbb Z} \Xi$ denote the free abelian group generated by the elements of $\Xi$. For
simplicity of notation we write $\Gamma ={\rm GL}_2({\mathbb Z})$. The following facts are well known
from the theory of Hecke operators. For any double coset $T_\alpha=\Gamma \alpha \Gamma$
in $\Xi$, there are finitely many $\alpha_i \in \Gamma \alpha \Gamma$ such that 
$\Gamma \alpha \Gamma=\sqcup_i \Gamma \alpha_i$. Thus, one can define a product
on $\Xi$ by setting 
\begin{equation}\label{HeckeOpsProd}
 T_\alpha T_\beta =\sum_\gamma c_{\alpha\beta}^\gamma T_\gamma 
\end{equation} 
where for $\Gamma \alpha \Gamma =\sqcup_i \Gamma \alpha_i$ and 
$\Gamma\beta \Gamma =\sqcup_j \Gamma \beta_j$, the coefficient 
$c_{\alpha\beta}^\gamma$ counts the number of pairs $(i,j)$ such that
$\Gamma \alpha_i\beta_j =\Gamma \gamma$. The ring structure on ${\mathbb Z}\Xi$
determined by the product \eqref{HeckeOpsProd} can equivalently be described
by considering finitely supported functions $f: \Xi \to {\mathbb Z}$ with the associative convolution product
\begin{equation}\label{HeckeConv}
(f_1\star f_2)(g) =\sum_h f_1(g h^{-1}) f_2(h) 
\end{equation}
where the sum is over the cosets $\Gamma h$ with $h\in {\rm GL}_2({\mathbb Q})$ or
equivalently over $\Gamma\backslash {\rm GL}_2({\mathbb Q})$.  The Hecke operators
are built in this form into the algebra of the ${\rm GL}_2$-system, through the
dependence on the variable $g\in \Gamma\backslash {\rm GL}_2({\mathbb Q})/\Gamma$,
see the discussion in \cite{CoMa-book}, Proposition~3.87.

\smallskip

We introduce here a variant ${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ of the ${\rm GL}_2$-algebra, where
an additional variable is introduced that accounts for the choice of the finite index
subgroup $G\subset {\rm GL}_2({\mathbb Z})$ through the coset spaces $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$,
for $\alpha \in {\rm GL}_2({\mathbb Q})$, which include for $\alpha=1$ the coset space $\P={\rm GL}_2({\mathbb Z})/G$.

\begin{lem}\label{cosetsHecke}
Let $G\subset {\rm GL}_2({\mathbb Z})$ be a finite index subgroup such that
$\alpha^{-1} G \alpha \cap {\rm GL}_2({\mathbb Z})$ is also a finite index subgroup, for all $\alpha \in {\rm GL}_2({\mathbb Q})$.
Consider the double coset ${\rm GL}_2({\mathbb Z}) \alpha G$, with the left action of ${\rm GL}_2({\mathbb Z})$ and the right action
of $G$.  The orbit spaces $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$ are finite. The algebra ${\mathbb Z}\Xi$ of Hecke 
operators acts on the module ${\mathbb Z} {\mathcal P}$ with ${\mathcal P}=\cup_\alpha \P_\alpha$. 
\end{lem}

\proof We show that the map ${\rm GL}_2({\mathbb Z})\to {\rm GL}_2({\mathbb Z}) \alpha G$ given by multiplication $\gamma \mapsto \gamma \alpha$ induces a bijection between ${\rm GL}_2({\mathbb Z})/(\alpha^{-1} G \alpha \cap {\rm GL}_2({\mathbb Z}))$ and ${\rm GL}_2({\mathbb Z}) \alpha G/G$, hence the orbit space $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$ is finite. 
For $\ell=\alpha^{-1} g \alpha \in \alpha^{-1} G \alpha \cap {\rm GL}_2({\mathbb Z})$ and
$\gamma \in {\rm GL}_2({\mathbb Z})$ we have $\gamma \ell \alpha  =\gamma \alpha g \sim \gamma \alpha$ in  
$\Gamma \alpha G/G$ so the map is well defined on equivalence classes. It is injective
since two $\gamma,\gamma'\in \Gamma$ with the same image differ by 
$\gamma'\gamma^{-1}=\alpha^{-1} \ell \alpha$ in $\alpha^{-1} G \alpha \cap \Gamma$ and it is also 
surjective by construction. The action of the algebra ${\mathbb Z}\Xi$ of Hecke operators is given by the
usual multiplication of cosets. We write $\Gamma \alpha \Gamma=\sqcup_i \Gamma \alpha_i$ for
finitely many $\alpha_i \in \Gamma \alpha \Gamma$ and $\Gamma \beta G=\sqcup_j \Gamma \beta_j$ for
finitely many $\beta_j \in \Gamma \beta G$. The product is then given by 
$$ \Gamma \alpha \Gamma \cdot \Gamma \beta G =\sum_\gamma c_{\alpha\beta}^\gamma \Gamma \gamma G$$ where $c_\gamma$ counts the number of pairs $(i,j)$ such that $\Gamma \alpha_i \beta_j =\Gamma \gamma$.
\endproof

\smallskip

The condition that $\alpha^{-1} G \alpha \cap {\rm GL}_2({\mathbb Z})$ is also a finite index subgroup, 
for all $\alpha \in {\rm GL}_2({\mathbb Q})$ is certainly satisfied, for instance, when $G$ is a congruence
subgroup. 

\smallskip

Given a generator $T_\alpha$ in the Hecke algebra ${\mathbb Z}\Xi$ and an element $\sum_s a_s \delta_s$
in ${\mathbb Z}{\mathcal P}$, we write the action of ${\mathbb Z}\Xi$ on the module ${\mathbb Z}{\mathcal P}$ as 
\begin{equation}\label{actionT1}
T_\alpha \sum_s a_s \delta_s = \sum_s a_s T_\alpha \delta_s =\sum_s a_s \sum_i c_{\alpha,s}^i \delta_{s_i} =
\sum_i (\sum_s a_s c_{\alpha,s}^i) \delta_{s_i}. 
\end{equation}
We can equivalently write elements of ${\mathbb Z}{\mathcal P}$ as finitely 
supported functions $\xi: {\mathcal P} \to {\mathbb Z}$, and elements of ${\mathbb Z}\Xi$ as finitely supported functions $f: \Xi \to {\mathbb Z}$,
and write the action 
\begin{equation}\label{actionT2}
(f \star \xi)(s)=\sum_h f(gh^{-1}) \xi(hs) 
\end{equation} 
where the sum is over cosets
$\Gamma h$ and for $\xi(s)=\sum_\sigma a_\sigma \delta_\sigma(s)$ we write $\xi(hs)$ for
\begin{equation}\label{actionT3}
 \xi(hs):=\sum_i (\sum_\sigma a_\sigma c_{h,\sigma}^i) \delta_{s_i}(s).
\end{equation} 

\smallskip

\begin{defn}\label{AcGP}
Let $G\subset {\rm GL}_2({\mathbb Z})$ be a finite index subgroup and let ${\mathcal P}=\cup_\alpha \P_\alpha$ with 
$\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$ for $\alpha \in {\rm GL}_2({\mathbb Q})$.
The involutive algebra ${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ is given by complex valued functions on the space
\begin{equation}\label{UpmGXi}
 {\mathcal U}^\pm_{G,{\mathcal P}} =\{ (g,\rho,z,\xi)  \in {\rm GL}_2({\mathbb Q})\times M_2(\hat {\mathbb Z}) \times \H^\pm \times {\mathcal P}
  \,|\, g\rho \in M_2(\hat {\mathbb Z})  \} ,
 \end{equation} 
that are invariant under the action of ${\rm GL}_2({\mathbb Z}) \times {\rm GL}_2({\mathbb Z})$ by $$(g,\rho,z,s) \mapsto (\gamma_1 g \gamma_2^{-1}, \gamma_2 z, \gamma_2 s). $$ 
Moreover, functions in ${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),\P}$ have finite support in ${\rm GL}_2({\mathbb Z})\backslash {\rm GL}_2({\mathbb Q})$ and in ${\mathcal P}$,
compact support in $z\in \H^\pm$, and they depend on the variable $\rho \in M_2(\hat {\mathbb Z})$ through the 
projection onto some finite level $p_N: M_2(\hat {\mathbb Z}) \to M_2({\mathbb Z}/N{\mathbb Z})$. 
The convolution product of ${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),{\mathcal P}}$ is given by
\begin{equation}\label{AGPconvol}
 (f_1\star f_2)(g,\rho,z,s) =\sum_{h \in {\rm GL}_2({\mathbb Z})\backslash {\rm GL}_2({\mathbb Q})\,:\, h\rho \in M_2(\hat{\mathbb Z})} f_1(g h^{-1}, h \rho, h(z), hs) f_2(h,\rho,z, s),
\end{equation} 
where we are using the notation \eqref{actionT2}, \eqref{actionT3} for the action of Hecke operators
on functions of ${\mathcal P}$.
The involution is $f^*(g,\rho,z,s)=\overline{f(g^{-1}, g\rho, g(z), g s)}$. The algebra
${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),\P}$ is endowed with a time evolution given by $$\sigma_t(f)(g,\rho,z,s) = 
|\det(g)|^{it} f(g,\rho,z,s).$$
\end{defn} 


\smallskip

We focus here on the algebra ${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ and we
construct Hilbert space representations analogous to the
ones considered for the original ${\rm GL}_2$-system. 

\smallskip

We consider the
set ${\mathcal S}_\rho:={\rm GL}_2({\mathbb Z})\backslash \{ g\in {\rm GL}_2({\mathbb Q}) \,:\, g\rho \in M_2(\hat{\mathbb Z})\}$. For
$\rho\in {\rm GL}_2(\hat{\mathbb Z})$ this set is ${\mathcal S}_\rho= {\rm GL}_2({\mathbb Z})\backslash M_2^\times({\mathbb Z})$,
with $M_2^\times({\mathbb Z})=\{ M\in M_2({\mathbb Z})\,|\, \det(M)\neq 0\}$. Consider then 
the Hilbert space ${\mathcal H}_\rho =\ell^2({\mathcal S}_\rho)$, and the representations
$\pi_{(\rho,z)}: {\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}} \to {\mathcal B}({\mathcal H}_\rho)$
$$ \pi_{(\rho,z)}(f) \xi (g) =\sum_{h\in {\mathcal S}_\rho} f(gh^{-1}, h\rho, h(z), hs) \xi(h). $$

We can complete the algebra ${\mathcal A}^c_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ to a $C^*$-algebra ${\mathcal A}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ 
in the norm $\| f \|=\sup_{(\rho,z,s)} \| \pi_{(\rho,z,s)}(f) \|_{{\mathcal B}({\mathcal H}_\rho)}$.
The time evolution is implemented in the representation $\pi_{(\rho,z,s)}$ by the Hamiltonian
$H \xi (g) = \log |\det(g)| \, \xi(g)$.

\smallskip

\subsection{The arithmetic algebra}\label{arithmSec}

We proceed exactly as in the case of the ${\rm GL}_2$-system of \cite{CoMa} to
construct an arithmetic algebra associated to ${\mathcal A}_{{\rm GL}_2({\mathbb Z}),\P}$. As in
\cite{CoMa} this will not be a subalgebra but an algebra of unbounded multipliers.

\smallskip

The arithmetic algebra ${\mathcal A}^{ar}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ is the algebra over ${\mathbb Q}$ obtained as follows.
We consider functions on ${\mathcal U}^\pm_{G,{\mathcal P}}$ of \eqref{UpmGXi} that are invariant under the action of ${\rm GL}_2({\mathbb Z}) \times {\rm GL}_2({\mathbb Z})$ by $(g,\rho,z) \mapsto (\gamma_1 g \gamma_2^{-1}, \gamma_2 \rho, \gamma_2 z)$ 
and that are finitely supported in $g\in G\backslash{\rm GL}_2({\mathbb Q})$ and in ${\mathcal P}$, 
that depend on $\rho$ through some finite level projection $p_N(\rho)\in M_2({\mathbb Z}/N{\mathbb Z})$ and that are
holomorphic in the variable $z\in \H$ and satisfy the growth condition that $| f(g,\rho,z,s) |$ 
is bounded by a polynomial in $\max\{ 1, \Im(z)^{-1} \}$ when $\Im(z)\to \infty$. The resulting algebra
${\mathcal A}^{ar}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ acts, via the convolution product \eqref{AGPconvol}, as unbounded
multipliers on the algebra ${\mathcal A}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$. This construction and its properties are
completely analogous to the original case of the  ${\rm GL}_2$-system and we refer the reader to
\cite{CoMa}, \cite{CoMa-book} for details.

\smallskip

The invariance
property ensures that these are modular functions for $G$ (written as ${\rm GL}_2({\mathbb Z})$-invariant 
functions on $\H\times \P$ rather than as $G$-invariant functions on $\H$). These functions
are endowed with the same convolution product \eqref{AGPconvol}. 

\medskip

\section{Boundary ${\rm GL}_2$-system}

We now consider how to extend this setting to incorporate the boundary $\P^1({\mathbb R})$
of the upper half-plane $\H$ and ${\mathbb Q}$-pseudo-lattices generalizing the ${\mathbb Q}$-lattices 
of \cite{CoMa}. A brief discussion of the boundary compactification of
the ${\rm GL}_2$-system was given in \S 7.9 of \cite{CoMa-book}, but the construction
of a suitable quantum statistical mechanical system associated to the boundary
was never worked out in detail. We introduce here an algebra associated to
the boundary $\P^1({\mathbb R})$ with the action of ${\rm GL}_2({\mathbb Z})$. Instead of considering
the full $\P^1({\mathbb R})$, we restrict to $[0,1]$, which meets every orbit of the
${\rm GL}_2({\mathbb Z})$ action, and we implement the action of the semigroup of
reduced matrices in ${\rm GL}_2({\mathbb Z})$ in the form of a crossed product algebra.
We analyze Hilbert space representations, time evolution, Hamiltonian,
partition function and KMS states.

\subsection{Boundary dynamics and symmetries}
 
The set $[0,1]\times \P$ meets every orbit of the action of ${\rm GL}_2({\mathbb Z})$ on $\P^1({\mathbb R})\times \P$,
acting on $\P^1({\mathbb R})$ by fractional linear transformations and on $\P= {\rm GL}_2({\mathbb Z})/G$ by the left-action
of ${\rm GL}_2({\mathbb Z})$ on itself. Moreover, two points $(x,s)$ and $(y,t)$ in $[0,1]\times \P$ are in the same
${\rm GL}_2({\mathbb Z})$-orbit iff there are integers $n,m\in {\mathbb N}$ such that $T^n(x,s)=T^m(y,t)$, where $T$ is the
shift operator on $[0,1]\times \P$ given by 
\begin{equation}\label{Tmap}
T: (x,s) \mapsto \left( \frac{1}{x}-\left[ \frac{1}{x} \right], \begin{pmatrix} -[1/x] & 1 \\ 1 & 0 \end{pmatrix} \cdot s \right). 
\end{equation}
 
\smallskip

\begin{lem}\label{actionP}
The action of the shift map \eqref{Tmap} on $[0,1]\times \P$ extends to an action on $[0,1]\times \P_\alpha$,
with $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$, for any given $\alpha \in {\rm GL}_2({\mathbb Q})$,
hence to an action on $[0,1]\times {\mathcal P}$ with ${\mathcal P}=\cup_\alpha \P_\alpha$.
\end{lem}

\proof The action of $T$ on $(x,s)\in [0,1]\times \P$ is implemented by the action of the matrix
$$ \begin{pmatrix}  -[1/x] & 1 \\ 1 & 0 \end{pmatrix} \in {\rm GL}_2({\mathbb Z}). $$
The same matrix acts by left multiplication on $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$, hence
it determines a map $T: [0,1]\times \P_\alpha \to [0,1]\times \P_\alpha$.
\endproof 

\smallskip

\begin{lem}\label{actionscommute}
Let ${\mathbb C}{\mathcal P}$ denote the ring of finitely supported complex valued functions on ${\mathcal P}$ and
let ${\mathcal B}={\mathcal B}([0,1],{\mathbb C}{\mathcal P})$ denote the ring of bounded functions from $[0,1]$ to ${\mathbb C}{\mathcal P}$.
Consider the action of the semigroup ${\mathbb Z}_+$ on ${\mathcal B}$ determined by 
the action of $T$ on $[0,1]\times {\mathcal P}$ of Lemma~\ref{actionP} and the action on ${\mathcal B}$
by Hecke operators acting on ${\mathbb C}{\mathcal P}$. These two actions commute.
\end{lem}

\proof We write functions $f \in {\mathcal B}$ in the form $\sum_\alpha f_\alpha(x,s_\alpha) \delta_\alpha$
where $\delta_\alpha$ is the characteristic function of $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$ and
$s_\alpha \in \P_\alpha$. The action of ${\mathbb Z}_+$ is given by 
$$ T^n: \sum_\alpha f_\alpha(x,s_\alpha)\delta_\alpha \mapsto \sum_\alpha f_\alpha(T^n(x,s_\alpha))\delta_\alpha, $$
with $T(x,s_\alpha)$ as in \eqref{Tmap}, while the action of a Hecke operator $T_\beta$ is given by
\begin{equation}\label{THeckeact}
 T_\beta: \sum_\alpha f_\alpha(x,s_\alpha)\delta_\alpha \mapsto  \sum_\gamma (\sum_\alpha c_{\beta,\alpha}^\gamma f_\alpha(x, s_\alpha)) \delta_\gamma, 
\end{equation} 
with $c_{\beta\alpha}^\gamma$ defined as in \eqref{actionT1}. It is then clear that these two actions commute.
\endproof

\smallskip

\begin{lem}\label{Tactiongk}
Let $X_k \subset [0,1]$ be the subset of points $x\in [0,1]$ with continued fraction expansion
starting with the digit $k\in {\mathbb N}$. Let ${\mathcal B}$ denote the algebra of bounded complex valued
functions on $[0,1] \times {\mathcal P}$. Let $\tau(f)=f\circ T$ denote the action of the 
shift $T: [0,1] \to [0,1]$ of the continued fraction expansion on $f\in {\mathcal B}$. 
Let ${\mathcal B}$ act as multiplication
operators on $L^2([0,1],d\mu)$ with $d\mu$ the unique $T$-invariant probability measure on $[0,1]$,
$$ d\mu(x)= \frac{1}{\log 2} \, \frac{1}{1+x} \, dx . $$
With the notation
\begin{equation}\label{gkmat}
g_k =\begin{pmatrix}  0 & 1 \\ 1 & k \end{pmatrix} \ \ \  \text{ with } \ \ \  g_k^{-1} = \begin{pmatrix} -k & 1 \\ 1 & 0 
\end{pmatrix},
\end{equation}
consider the operators 
\begin{equation}\label{Skops}
S_k \xi (x) =\chi_{X_k}(x) \cdot \xi (g_k^{-1} x) \,  \ \ \ \text{ and } \ \ \  \tilde S_k \xi (x) = \xi( g_k\, x ),
\end{equation}
for $\xi\in L^2([0,1],d\mu)$, with $\chi_{X_k}$ the characteristic function of the subset $X_k\subset [0,1]$.
These satisfy $\tilde S_k=S_k^*$ with $S_k^* S_k =1$ and $\sum_k S_k S_k^*=1$
They also satisfy the relation 
\begin{equation}\label{TSk}
\sum_k S_k \, f \, S_k^* =f\circ T .
\end{equation}
\end{lem}

\proof
The shift map of the continued fraction expansion, given by $T(x) =1/x - [1/x]$, 
acts on $x\in X_k$ as $x\mapsto g_k^{-1} x$ with the matrix $g_k^{-1}$ acting by fractional linear transformations.
The operators $S_k$ defined as in \eqref{Skops} are not isometries
on $L^2([0,1],dx)$ with respect to the Lebesgue measure $dx$. However, if we consider the
unique $T$-invariant probability measure on $[0,1]$ given by $d\mu(x)=\log(2)^{-1}(1+x)^{-1} dx$, then we have
$d\mu\circ g_k^{-1}|_{X_k}=d\mu |_{X_k}$ for all $k\in {\mathbb N}$, hence
$\langle S_k \xi_1, S_k \xi_2\rangle =\int_{X_k} \bar \xi_1\circ g_k^{-1}\, \xi_2\circ g_k^{-1} \, d\mu=
\int_{X_k} \bar \xi_1\circ g_k^{-1}\, \xi_2\circ g_k^{-1} \, d\mu\circ g_k^{-1} =\int_{[0,1]} \bar \xi_1\, \xi_2\, d\mu=
\langle \xi_1, \xi_2\rangle$. We have $\tilde S_k\, S_k \xi(x) = \xi(x) \chi_{X_k}(g_k x)= \xi(x)$.
Moreover, $\tilde S_k=S_k^*$ in this inner product since we have
$\langle \xi_1, S_k \xi_2\rangle =\int_{X_k} \bar \xi_1\, \xi_2\circ g_k^{-1} \, d\mu =
\int_{X_k} \bar \xi_1\, \xi_2\circ g_k^{-1} \, d\mu\circ g_k^{-1} = \int_{[0,1]} \bar \xi_1\circ g_k\, \xi_2 \, d\mu =
\langle \tilde S_k \xi_1, \xi_2 \rangle$.
We also have $\sum_k S_k \, f \, S_k^* \xi(x) = \sum_k f(g_k^{-1} x)  \chi_{X_k}(x) \xi(x) =f(T(x))$.
Thus we obtain $\sum_k S_k \, f \, S_k^* =f\circ T$, which in particular also implies 
$\sum_k S_k \, S_k^* =1$. 
\endproof

\smallskip

Consider the set of matrices in ${\rm GL}_2({\mathbb Z})$ of the form
\begin{equation}\label{Redn}
{\rm Red}_n:=\left\{ \,
\begin{pmatrix}
0 & 1\\
1 & k_1
\end{pmatrix} \ \dots \
\begin{pmatrix}
0 & 1\\
1 & k_n
\end{pmatrix}\,|\,k_1,\dots , k_n\ge 1;\, k_i\in{\mathbb Z}\right\} 
\end{equation}
The semigroup of reduced matrices is defined as
\begin{equation}\label{Red}
 {\rm Red}:=\cup_{n\ge 1}\, {\rm Red}_n \subset {\rm GL}_2({\mathbb Z}) 
\end{equation} 
An equivalent description of the ${\rm Red}$ semigroup is given by (\cite{LewZag})
$$ {\rm Red}=\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in {\rm GL}_2({\mathbb Z}) \,|\,  0\leq a \leq b, \, 0\leq c\leq d \}. $$

\smallskip

\begin{lem}\label{homRed}
Assigning to a matrix
\begin{equation}\label{prodRed}
 \gamma = \begin{pmatrix} 0 & 1 \\ 1 & n_1 \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ 1 & n_N \end{pmatrix} 
\end{equation} 
in ${\rm Red}$ the product $n_1\cdots n_N \in {\mathbb N}$ is a well defined semigroup homomorphism.
\end{lem}

\proof We only need to check that the representation of a matrix $\gamma$ in ${\rm Red}$ as
a product \eqref{prodRed} is unique so that the map is well defined. It is then by construction a
semigroup homomorphism. The group ${\rm GL}_2({\mathbb Z})$ has generators 
$$ \sigma=\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \ \ \ \  \rho=\begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} $$
with relations $(\sigma^{-1}\rho)^2 =(\sigma^{-2}\rho^2)^6 =1$. The semigroup ${\rm Red}$ can be
equivalently described as the subsemigroup of the semigroup generated by $\sigma$ and $\rho$
made of all the words in $\sigma$, $\rho$ that end in $\rho$, so 
elements are products of matrices of the form $\sigma^{n-1}\rho =\begin{pmatrix} 0 & 1 \\ 1 & n \end{pmatrix}$,
$$ \gamma = \begin{pmatrix} 0 & 1 \\ 1 & n_1 \end{pmatrix} \cdots \begin{pmatrix} 0 & 1 \\ 1 & n_{\ell(\gamma)} \end{pmatrix} $$
where $\ell(\gamma)$ to be the number of $\rho$'s in the word in $\sigma$ and $\rho$
representing $\gamma$.  The semigroup generated by $\sigma$ and $\rho$ is a free semigroup, 
as the only relations in ${\rm GL}_2({\mathbb Z})$ between these generators involve the inverse $\sigma^{-1}$. 
In an element $\gamma \in {\rm Red}$ had two different representations \eqref{prodRed}, for two
different ordered sets $\{ n_1, \ldots, n_N \}$ and $\{ k_1, \ldots, k_L \}$ then we would have a
relation $\sigma^{n_1-1}\rho \sigma^{n_2-1}\rho \cdots \sigma^{n_N-1}\rho =
\sigma^{k_1-1}\rho \sigma^{k_2-1}\rho\cdots \sigma^{k_L-1}\rho$ involving the generators
$\sigma$ and $\rho$ but not their inverses, which would contradict the fact that $\sigma$ and
$\rho$ generate a free semigroup.
\endproof 

\smallskip

\begin{lem}\label{lemRedHecke}
Let ${\mathcal B}$ denote the algebra of bounded complex valued functions on $[0,1] \times {\mathcal P}$
that are finitely supported in ${\mathcal P}$.
The transformations $\alpha_\gamma(f)=\chi_{X_\gamma} \cdot f\circ \gamma^{-1}$ define 
a semigroup action of ${\rm Red}$ on ${\mathcal B}$. This action commutes with the
action of Hecke operators.
\end{lem}

\proof We check that $\alpha_\gamma(f)=\chi_{X_\gamma} \cdot f\circ \gamma^{-1}$ is 
a well defined semigroup action of ${\rm Red}$ on ${\mathcal B}$. For $\gamma$ of the form 
\eqref{prodRed} we have $\alpha_\gamma=\alpha_{g_1}\cdots \alpha_{g_n}$ with the
factors $g_k$ as in \eqref{gkmat}, since for two matrices $\gamma,\gamma'$ in ${\rm Red}$ related
by $\gamma' =g_k \gamma$ for some $g_k$ as in \eqref{gkmat} we have
$\chi_{X_k} \cdot \chi_{X_{\gamma'}}\circ g_k^{-1}=\chi_{X_\gamma}$. The
commutation with the action of Hecke operators can be checked as in the case
of the shift $T$ in Lemma~\ref{actionscommute}. We write elements of the algebra
in the form $\sum_\alpha f_\alpha(x,s_\alpha) \delta_\alpha$
where $\delta_\alpha$ is the characteristic function of $\P_\alpha={\rm GL}_2({\mathbb Z}) \alpha G/G$ and
$s_\alpha \in \P_\alpha$, with the action of Hecke operators as in \eqref{THeckeact}. 
The action of $\gamma \in {\rm Red}$ on the other hand is given by
$\alpha_{\gamma}\,  \sum_\alpha f_\alpha(x,s_\alpha)\delta_\alpha=\sum_\alpha \chi_{X_\gamma}(x)\, 
f(\gamma^{-1}(x,s_\alpha)) \delta_\alpha$. These actions commute, as in the case of 
Lemma~\ref{actionscommute}.
\endproof


\smallskip
\subsection{A boundary algebra}

We now introduce an algebra associated to the boundary of the ${\rm GL}_2$-system.
Instead of requiring invariance with respect to the ${\rm GL}_2({\mathbb Z})$
action in the boundary variable $\theta \in \P^1({\mathbb R})$, we restrict the variable to the
interval $[0,1]$, because of the prior observation  that this set meets every ${\rm GL}_2({\mathbb Z})$-orbit
and we implement the action of the shift operator $T$ in the form of a semigroup crossed
product algebra. This corresponds to taking the quotient by the action of $T$ (hence by the
action of ${\rm GL}_2({\mathbb Z})$) in a noncommutative way, by considering a crossed product
algebra instead of an algebra of functions constant along the orbits. This will be a
semigroup crossed product with respect to the semigroup ${\rm Red}$ discussed
above, and in a form that will implement the action of the shift operator $T$ as 
in Lemma~\ref{Tactiongk}. The semigroup crossed product approach is better
in the boundary case than requiring ${\rm GL}_2({\mathbb Z})$-invariance as in the bulk case, 
because, unlike the bulk case, the boundary action of ${\rm GL}_2({\mathbb Z})$ on $\P^1({\mathbb R})$ has dense orbits. 
We will also be considering bounded functions
of $x\in [0,1]$, rather than continuous. The reason will be clear when we consider
such boundary functions that are obtained as integration on certain
configurations of geodesics in the bulk space.

\smallskip

\begin{defn}\label{subalg0}
Let ${\mathcal A}_\partial^c$ denote the associative algebra of continuous complex valued functions on
\begin{equation}\label{UG0}
 {\mathcal U}_{\partial,G}=\{ (g, \rho, s) \in {\rm GL}_2({\mathbb Q}) \times M_2(\hat {\mathbb Z})\times {\mathcal P}  \,|\, g\rho\in M_2(\hat {\mathbb Z})  \} 
\end{equation}
that are invariant with respect to the action of $\Gamma\times \Gamma$ by
$(g,\rho) \mapsto (\gamma_1 g \gamma_2^{-1}, \gamma_2\rho)$ and are finitely
supported in ${\mathcal P}$ and in ${\rm GL}_2({\mathbb Z}) \backslash {\rm GL}_2({\mathbb Q})$ with the dependence
on $\rho$ through a finite level projection $p_N(\rho)\in M_2({\mathbb Z}/N{\mathbb Z})$, endowed with the
convolution product
\begin{equation}\label{convol0}
(f_1\star f_2) (g, \rho, s) = \sum_{h\in \Gamma\backslash {\rm GL}_2({\mathbb Q})\,:\, h\rho\in M_2(\hat{\mathbb Z})} 
f_1(gh^{-1}, h\rho, hs) f_2(h,\rho,s) 
\end{equation}
and with the involution $f^*(g, \rho, s) =\overline{f(g^{-1},g\rho,gs)}$. Let 
$\pi_{\rho,s}: {\mathcal A}_\partial^c \to {\mathcal B}({\mathcal H}_\rho)$ be the representation 
$\pi_{\rho,s}(f)\xi(g)=\sum_h f(gh^{-1},h\rho,hs) \xi(h)$ for $h\in \Gamma\backslash G_\rho$
with $G_\rho=\{ g\in {\rm GL}_2({\mathbb Q})\,|\, g\rho\in M_2(\hat{\mathbb Z})\}$. 
Let ${\mathcal A}_\partial$ denote the $C^*$-algebra
completion of ${\mathcal A}_\partial^c$ with respect to $\| f \| =\sup \| \pi_{\rho,s}(f) \|_{{\mathcal H}_\rho}$.
Let ${\mathcal B}_\partial ={\mathcal B}([0,1],{\mathcal A}_\partial)$ be the algebra of bounded functions from $[0,1]$ to ${\mathcal A}_\partial$,
with pointwise product
$$ (f_1\star f_2)(g,\rho,x,s)=\sum_h f_1(gh^{-1},h\rho,x,hs) f_2(h,\rho,x,s) $$
and involution $f^*(g,\rho,x,s)=\overline{f(g^{-1},g\rho,x,gs)}$.
\end{defn}

\smallskip

\begin{defn}\label{semicrossdef}
Let ${\mathcal A}_{\partial, G,{\mathcal P}}$ be the involutive associative algebra generated by ${\mathcal B}_\partial$ and by
isometries $S_k$, $k\geq 1$ with $S_k^* S_k=1$ and $\sum_{k\in {\mathbb N}} S_k S_k^* =1$ and satisfying the relations
\begin{equation}\label{relSkgk}
S_k\, f =\chi_{X_k}\, \cdot\, f\circ g_k^{-1}\, \cdot \,S_k \ \ \ \text{ and }  \ \ \  S_k^*\, f = f\circ g_k \, \cdot \, S_k^*,
\end{equation}
where $\chi_{X_k}$ is the characteristic function of the subset $X_k\subset [0,1]$ of points with
continued fraction expansion starting with $k$, and the matrices $g_k, g_k^{-1}$ in ${\rm GL}_2({\mathbb Z})$ are as
in \eqref{gkmat}, and also the relation
\begin{equation}\label{relTSks}
\sum_{k\in {\mathbb N}} S_k f S_k^* = f \circ T 
\end{equation}
for all $f\in B_\partial$, where for $f=f(g,\rho,x,s)$ we have $(f\circ T)(g,\rho,x,s)=f(g,\rho,T(x,s))$,
with the action of $T$ on $[0,1]\times {\mathcal P}$ as in Lemma~\ref{actionP}, and with $f\circ g_k^{\pm 1}(g,\rho,x,s)=
f(g,\rho,g_k^{\pm 1} (x,s))$.
The involution on ${\mathcal A}_{\partial, G,{\mathcal P}}$ is given by the involution on ${\mathcal B}_\partial$ and by $S_k  \mapsto S_k^*$.
\end{defn}

In fact, the relation \eqref{relTSks} follows from the relations \eqref{relSkgk} as in Lemma~\ref{Tactiongk},
but we write it explicitly as it is the relation that implements the dynamical system $T$. Note that we
are implicitly using in the construction of the algebra the fact that the semigroup action of ${\rm Red}$
and the action of Hecke operators (that is built into the convolution product of ${\mathcal B}_\partial$
commute as in Lemma~\ref{lemRedHecke}. 

\smallskip

\begin{lem}\label{crossRed}
The algebra ${\mathcal A}_{\partial, G,{\mathcal P}}$ can be identified with the semigroup crossed product
${\mathcal B}_\partial \rtimes {\rm Red}$ of the algebra ${\mathcal B}_\partial$ of Definition~\ref{subalg0} and
the semigroup of reducible matrices, with respect to the action $\alpha: {\rm Red} \to {\rm Aut}(B_\partial)$
by $\alpha_\gamma(f)=\chi_{X_\gamma} \cdot f\circ \gamma^{-1}$, where for $\gamma$ of the form 
\eqref{prodRed}, the set $X_\gamma \subset [0,1]$ is the cylinder set consisting of points with
continued fraction expansion starting with the sequence $n_1,\ldots, n_N$.
\end{lem}

\proof We see as in Lemma~\ref{lemRedHecke} that 
$\alpha_\gamma(f)=\chi_{X_\gamma} \cdot f\circ \gamma^{-1}$ defines a
semigroup action of ${\rm Red}$ on ${\mathcal B}_\partial$.
The semigroup crossed product
algebra is generated by $B_\partial$ and by isometries $\mu_\gamma$ for $\gamma\in {\rm Red}$
satisfying $\mu_\gamma \mu_{\gamma'}=\mu_{\gamma \gamma'}$ for all $\gamma,\gamma' \in {\rm Red}$,
$\mu_\gamma^* \mu_\gamma=1$ for all $\gamma\in {\rm Red}$ and 
$\mu_\gamma\, f\, \mu_\gamma^* =\alpha_\gamma(f)$.  It suffices to consider isometries $\mu_{g_k} =:S_k$
associated to the elements $g_k \in {\rm Red}$ as in \eqref{gkmat} with $\mu_\gamma =S_{n_1}\cdots S_{n_N}$
for $\gamma\in {\rm Red}$ as in \eqref{prodRed}. We then see that the generators and relations of the
algebras $B_\partial \rtimes {\rm Red}$ agree with those of the algebra ${\mathcal A}_{\partial, G,{\mathcal P}}$ of
Definition~\ref{semicrossdef}.
\endproof

\smallskip

\begin{cor}\label{BEsubalg}
Let $E\subset [0,1]\times {\mathcal P}$ be a subset invariant under the action of the shift $T$ of
the continued fraction expansion. Then $E$ determines an algebra of ${\mathcal A}_{\partial, G,{\mathcal P},E}$ 
given by $B_{\partial,E}\rtimes {\rm Red}$ where $B_{\partial,E}={\mathcal B}(E,{\mathcal A}_\partial)$ is the algebra 
of bounded functions from $E$ to ${\mathcal A}_\partial$ as in Definition~\ref{subalg0}. Extending the
functions by zero outside of $E$ realizes ${\mathcal A}_{\partial, G,{\mathcal P},E}$ as a subalgebra of
${\mathcal A}_{\partial, G,{\mathcal P}}$.
\end{cor}

\proof If the set $E$ is $T$-invariant then the algebra $B_{\partial,E}$ is
invariant under the action of the semigroup ${\rm Red}$ by 
$\alpha_\gamma(f)=\chi_{X_\gamma} \cdot f\circ \gamma^{-1}$, since for
$\gamma=g_{k_1}\cdots g_{k_n}$, the matrix $\gamma^{-1}$ acts on
$X_\gamma$ as the shift $T^n$. Thus, we can form the semigroup
crossed product algebra $B_{\partial,E}\rtimes {\rm Red}$ as in 
Lemma~\ref{crossRed}. Since $B_{\partial,E}$ and $B_\partial$ are
algebras of bounded functions that are not required to be continuous,
extending the functions to zero outside of $E$ realizes 
$B_{\partial,E}$ as a subalgebra of $B_\partial$ compatibly with the
action of the semigroup ${\rm Red}$.
\endproof



\smallskip
\subsection{Representations and time evolution}

Let ${\mathcal H}_\rho$ be the same Hilbert space considered above, ${\mathcal H}_\rho=\ell^2({\mathcal S}_\rho)$ with
${\mathcal S}_\rho={\rm GL}_2({\mathbb Z})\backslash G_\rho$ where $G_\rho=\{ g\in {\rm GL}_2({\mathbb Q}) \,|\, g\rho \in M_2(\hat{\mathbb Z}) \}$.
We will consider the case of an invertible $\rho\in {\rm GL}_2(\hat{\mathbb Z})$, for which 
${\mathcal S}_\rho ={\rm GL}_2({\mathbb Z})\backslash M_2^\times({\mathbb Z})$, with $M_2^\times({\mathbb Z})=\{ M \in M_2({\mathbb Z})\,|\, \det(M)\neq 0 \}$.
Let ${\mathcal W}=\cup_N {\mathcal W}_N$ denote the set of all finite sequences $k_1,\ldots, k_N$ with
$k_i \in {\mathbb N}$, including an element $\emptyset$ corresponding to the empty sequence. 
Consider the Hilbert spaces $\ell^2({\mathcal W})$ and $\tilde{\mathcal H}_\rho =\ell^2({\mathcal W})\otimes {\mathcal H}_\rho$.

\smallskip

\begin{lem}\label{lemRepCross}
The algebra ${\mathcal A}_{\partial, G,{\mathcal P}}={\mathcal B}_\partial \rtimes {\rm Red}$ acts on the Hilbert space
$\tilde{\mathcal H}_\rho$ through the representations
\begin{equation}\label{repsCross}
\pi_{\rho,x,s}(f)\,  (\xi(g)\otimes \epsilon_{k_1,\ldots, k_N}) = \sum_{h\in {\mathcal S}_\rho} 
f(g h^{-1}, h\rho, g_\gamma (x, hs)) \, \xi(h)\otimes \epsilon_{k_1,\ldots, k_N} 
\end{equation}
for $f\in B_\partial$ and with $g_\gamma=g_{k_1}\cdots g_{k_N}$, with $g_{k_i}$ as in \eqref{gkmat}, and 
\begin{equation}\label{repsCrossS}
S_k\, (\xi\otimes\epsilon_{k_1,\ldots, k_N}) = \xi\otimes \epsilon_{k, k_1,\ldots, k_N},\ \ \ \text{ and } \ \ \  
S_k^* (\xi\otimes \epsilon_{k_1,\ldots, k_N}) = \left\{ \begin{array}{ll} \xi\otimes \epsilon_{k_2,\ldots, k_N} & k_1=k \\
0 & \text{otherwise.} \end{array}\right.
\end{equation}
\end{lem}

\proof We check that \eqref{repsCross} gives a representation of the subalgebra $B_\partial$ and that
the operators $\pi_{\rho,x,s}(f)$, $S_k$, $S^*_k$ of \eqref{repsCross}  and \eqref{repsCrossS} satisfy
the relations $S_k^* S_k=1$, $\sum_k S_k S_k^*=1$, 
$S_k \pi_{\rho,x,s}(f)=\pi_{\rho,x,s}(\chi_{X_k} \, f\circ g_k^{-1}) S_k$ and 
$S_k^* \pi_{\rho,x,s}(f)=\pi_{\rho,x,s}(f\circ g_k) S_k^*$. For the first property it suffices to see
that $\pi_{\rho,x,s}(f_1 \star f_2)=\pi_{\rho,x,s}(f_1) \circ \pi_{\rho,x,s}(f_2)$. We have
$$ \pi_{\rho,x,s}(f_1 \star f_2) \, (\xi(g)\otimes \epsilon_{k_1,\ldots, k_N}) =\sum_{h\in {\mathcal S}_\rho} 
(f_1\star f_2)(g h^{-1}, h\rho, g_\gamma (x, hs)) \, \xi(h)\otimes \epsilon_{k_1,\ldots, k_N}  = $$
$$ \sum_{h\in {\mathcal S}_\rho} \sum_{\ell \in {\mathcal S}_\rho} f_1(g h^{-1} \ell^{-1}, \ell h \rho, g_\gamma (x, \ell h s))
f_2(\ell, h \rho, g_\gamma (x, hs)) \xi(h)\otimes \epsilon_{k_1,\ldots, k_N}, $$
where we used Lemma~\ref{lemRedHecke}. This is then equal to
$$ \sum_{\ell \in {\mathcal S}_\rho} f_1(g h^{-1} \ell^{-1}, \ell h \rho, g_\gamma (x, \ell h s)) (\pi_{\rho,x,s}(f_2) \xi) (\ell) \otimes \epsilon_{k_1,\ldots, k_N} = \pi_{\rho,x,s}(f_1) \pi_{\rho,x,s}(f_2) \xi \otimes \epsilon_{k_1,\ldots, k_N}.  $$
The relations $S_k^* S_k=1$ and $\sum_k S_k S_k^*=1$ follow directly from \eqref{repsCrossS}. For 
relations between the $S_k,S_k^*$ and the $\pi_{\rho,x,s}(f)$, we have
$$ S_k \pi_{\rho,x,s}(f) \, \xi \otimes \epsilon_{k_1,\ldots, k_N} = S_k
\sum_{h\in {\mathcal S}_\rho} f(g h^{-1}, h\rho, g_k^{-1} g_k g_\gamma (x, hs)) \, \xi(h)\otimes \epsilon_{k_1,\ldots, k_N} $$
$$ =  
\sum_{h\in {\mathcal S}_\rho} f(g h^{-1}, h\rho, g_k^{-1} g_k g_\gamma (x, hs))\, \chi_{X_k}(g_k g_\gamma x) \, \xi(h)\otimes \epsilon_{k, k_1,\ldots, k_N}, $$
for $g_\gamma=g_{k_1}\cdots g_{k_N}$, with $\chi_{X_k}(g_k g_\gamma x)=1$, so we get
$$ \pi_{\rho,x,s}(\chi_{X_k}\cdot f\circ g_k^{-1}) \, S_k \, \xi \otimes \epsilon_{k, k_1,\ldots, k_N}. $$
The second relation is similar: we have
$$ S_k^* \pi_{\rho,x,s}(f) \, \xi \otimes \epsilon_{k_1,\ldots, k_N} = S_k^* 
\sum_{h\in {\mathcal S}_\rho} f(g h^{-1}, h\rho, g_k^{-1} g_k g_\gamma (x, hs)) \, \xi(h)\otimes \epsilon_{k_1,\ldots, k_N} $$
$$ =  \sum_{h\in {\mathcal S}_\rho} f(g h^{-1}, h\rho, g_k g_{\gamma'} (x, hs)) \, S_k^* \xi(h)\otimes 
\epsilon_{k_1,\ldots, k_N},  $$
with $g_{\gamma'}=g_{k_2}\cdots g_{k_n}$, so that we obtain
$$ \pi_{\rho,x,s}(f\circ g_k)\, S_k^*\, \xi \otimes \epsilon_{k_1,\ldots, k_N}. $$
Thus, \eqref{repsCross}  and \eqref{repsCrossS} determine a representation of ${\mathcal A}_{\partial, G,{\mathcal P}}={\mathcal B}_\partial \rtimes {\rm Red}$ by bounded operators on the Hilbert space $\tilde{\mathcal H}_\rho$. 
\endproof

\smallskip

\begin{prop}\label{timeevolZ}
The transformations $\sigma_t(f)(g,\rho,x,s)=|\det(g)|^{it} f(g,\rho,x,s)$ and $\sigma_t(S_k)=k^{it} S_k$
define a time evolution $\sigma: {\mathbb R} \to {\rm Aut}({\mathcal A}_{\partial, G,{\mathcal P}})$. In the representations of Lemma~\ref{lemRepCross} on $\tilde{\mathcal H}_\rho$ with $\rho\in {\rm GL}_2(\hat{\mathbb Z})$ 
the time evolution is implemented by the Hamiltonian
\begin{equation}\label{Hsigmat}
H\, \xi(g)\otimes \epsilon_{k_1,\ldots, k_n}= \log (|\det(g)| \cdot k_1 \cdots k_n) \, 
\xi(g)\otimes \epsilon_{k_1,\ldots, k_n},
\end{equation}
with partition function
\begin{equation}\label{Zetabeta}
Z(\beta)={\rm Tr}(e^{-\beta H}) =\frac{\zeta(\beta) \zeta(\beta-1)}{2-\zeta(\beta)}  
\end{equation}
with $\zeta(\beta)$ the Riemann zeta function.
\end{prop}

\proof We have $$\sigma_t(f_1\star f_2)(g,\rho,x,s)=\sigma_t (\sum_h f_1(gh^{-1},h\rho,x,hs) f_2(h,\rho,x,s))= $$ $$
\sum_h |\det(gh^{-1}|^{it} |\det(h)|^{it} f_1(gh^{-1},h\rho,x,hs) f_2(h,\rho,x,s)=\sigma_t(f_1)\star \sigma_t(f_2).$$
We also have $\sigma_t(S_k^*)=k^{-it} S_k^*$ and we see that the action of
$\sigma_t$ is compatible with the relations in ${\mathcal A}_{\partial, G,{\mathcal P}}$ and defines a $1$-parameter family
of algebra homomorphisms. By direct comparison between \eqref{Hsigmat} and \eqref{repsCross}  and \eqref{repsCrossS} we also see that 
$$ \pi_{\rho,x,s}(\sigma_t(f))= e^{it H}\, \pi_{\rho,x,s}(f) \, e^{-itH} \ \ \  \text{ and } \ \ \ 
\sigma_t(S_k) = e^{it H}\, S_k e^{-it H}\, .$$
We have
$$ Z(\beta)=\sum_{g\in {\mathcal S}_\rho} |\det(g)|^{-\beta} \cdot \sum_{k=k_1\cdots k_n} k^{-\beta}. $$
For the first sum, for $\rho\in {\rm GL}_2(\hat{\mathbb Z})$ we have ${\mathcal S}_\rho={\rm GL}_2({\mathbb Z})\backslash M_2^\times({\mathbb Z})$,
where $M_2^\times({\mathbb Z})=\{ M \in M_2({\mathbb Z}) \,|\, \det(M)\neq 0 \}$.
Thus, we are counting $\{ M\in M_2^\times({\mathbb Z})\,|\, |\det(M)|=n \}$ modulo ${\rm GL}_2({\mathbb Z})$. Up to a change
of basis in ${\rm GL}_2({\mathbb Z})$ we can always write a sublattice of ${\mathbb Z}^2$ in the form
$$ \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} {\mathbb Z}^2 $$
with $a,d\geq 1$ and $0\leq b < d$, \cite{Serre}. 
Thus, we are equivalently counting such matrices with determinant $n$.
This counting is given by $\sigma_1(n)=\sum_{d|n} d$ so the first sum is
$\sum_{n\geq 1} \sigma_1(n)\, n^{-\beta} =\zeta(\beta)\zeta(\beta-1)$ as in the original ${\rm GL}_2$-system.
To compute the second sum, let $P_n$ denote the total number of ordered factorizations of
$n$ into nontrivial positive integer factors. The sum we are considering is
$$ \sum_{n\geq 1} P_n \, n^{-\beta} = 1 + \sum_{k=1}^\infty \sum_{n=2}^\infty n^{-\beta} \sum_{n=n_1\cdots n_k} 1 $$
$$ = 1 + \sum_{k\geq 1} \prod_{i=1}^k (\sum_{n_i\geq 2} n_i^{-\beta}) $$ $$ = \sum_{k=0}^\infty (\zeta(\beta)-1)^k =
\frac{1}{2-\zeta(\beta)}. $$
\endproof

\smallskip

For the purpose of this paper we focus on the low temperature Gibbs states of the system
$({\mathcal A}_{\partial,G,{\mathcal P}},\sigma_t)$. A more detailed analysis of the KMS states will be 
discussed separately. 

\begin{lem}\label{ZbetaPoles}
The partition function $Z(\beta)$ of Proposition~\ref{timeevolZ} is defined by an absolutely
convergent series $Z(\beta)={\rm Tr}(e^{-\beta H})=\sum_{\lambda\in {\rm Spec}(H)} e^{-\beta \lambda}$,
for $\beta >2$. Its analytic continuation \eqref{Zetabeta} has poles at $\beta\in \{ 1, \beta_c , 2 \}$,
for a point $\beta_c \sim 1.728647$.
\end{lem}

\proof The Riemann zeta function $\zeta(\beta)$ has a pole at $\beta=1$ and in 
the range $\beta \in (1,\infty)$ it is decreasing to $1$ and it crosses the value $\zeta(\beta)=2$
at a point  $\beta_c \sim 1.728647$. The sum $\sum_{n\geq 1} \sigma_1(n) n^{-\beta}$ is
convergent for $\beta>2$ and its analytic continuation $\zeta(\beta)\zeta(\beta-1)$ has
poles at $\beta=2$ and $\beta=1$. 
\endproof  

Thus, the low temperature Gibbs states we will be considering are states in the
range $\beta >2$ where the operator $e^{-\beta H}$ is trace class, of the form, for $X\in {\mathcal A}_{\partial,G,{\mathcal P}}$
$$ \varphi_\beta( X ) =\frac{{\rm Tr}( X e^{-\beta H })}{{\rm Tr}(e^{-\beta H}) }= Z(\beta)^{-1} \sum_{k_i, h} |\det(h)|^{-\beta} (k_1\cdots k_n)^{-\beta}\,  \langle \delta_h \otimes \epsilon_{k_1,\ldots, k_n}, X \delta_h \otimes \epsilon_{k_1,\ldots, k_n}\rangle. $$
On the subalgebra $X=f(g,\rho,x,s) \in B_\partial$, these Gibbs states give
$$ \varphi_{\beta,\rho,x,s}(f)=Z(\beta)^{-1} \sum_{k_i, g} |\det(g)|^{-\beta} (k_1\cdots k_n)^{-\beta} f(1,g\rho, g_\gamma(x,gs)), $$
for $\gamma \in {\rm Red}$ determined by the sequence $k_1, \ldots, k_n$. 

\smallskip

As in \cite{CoMa} we consider the ground states at zero temperature as the
weak limit of the Gibbs states for $\beta\to \infty$
$$ \varphi_{\infty,\rho,x,s}(f) =\lim_{\beta \to \infty} \varphi_{\beta,\rho,x,s}(f). $$
These give the projection onto the kernel of the Hamiltonian
$$ \varphi_{\infty,\rho,x,s}(f) =\langle \delta_1 \otimes \epsilon_\emptyset, \pi_{\rho,x,s}(f)\, \delta_1 \otimes \epsilon_\emptyset\rangle = f(1,\rho,x,s), $$
the evaluation of the function $f$ at the point $g=1$ and $(\rho,x,s)$ that determines the representation
$\pi_{\rho,x,s}$. 
 

\medskip
\section{Averaging on geodesics and boundary values}

In this section we construct boundary values of the observables of the
${\rm GL}_2$-system of \S \ref{GL2Sec}. We use the theory of limiting modular
symbols of  \cite{ManMar}. We show that the resulting boundary values are localized at the
quadratic irrationalities.


\subsection{Geodesics between cusps}

Let $C_{\alpha,\epsilon,s}$ with $\alpha \in \P^1({\mathbb Q})\smallsetminus \{ 0 \}$, $\epsilon\in \{ \pm \}$,
and $s\in \P$ denote the geodesic in $\H^\epsilon \times \P$ with endpoints at the cusps 
$(0,s)$ and $(\alpha,s)$ in $\P^1({\mathbb Q})\times \P$. 

\smallskip

Let $p_k(\alpha), q_k(\alpha)$ be the successive numerators and denominators of the 
${\rm GL}_2({\mathbb Z})$-continued fraction expansion of $\alpha\in {\mathbb Q}$ with $p_n(\alpha)/q_n(\alpha)=\alpha$ and let
$$ g_k(\alpha):=\begin{pmatrix} p_{k-1}(\alpha) & p_k(\alpha) \\ q_{k-1}(\alpha) & q_k(\alpha) \end{pmatrix} \in {\rm GL}_2({\mathbb Z}). $$ 

\smallskip

We denote by $C^k_{\alpha,\epsilon,s}$ the geodesic in $\H^\epsilon \times \P$ with endpoints
at the cusps 
$$ \frac{p_{k-1}(\alpha)}{q_{k-1}(\alpha)} = g_k^{-1}(\alpha)\cdot 0 \ \ \text{ and } \ \ 
\frac{p_k(\alpha)}{q_k(\alpha)} = g_k^{-1}(\alpha)\cdot \infty , $$
where $g\cdot z$ for $g\in {\rm GL}_2({\mathbb Z})$ and $z\in \H^\pm$ is the action by fractional linear transformations.

\smallskip

For $C$ a geodesic in $\H^\pm$ let $ds_C$ denote the geodesic length element. In the case of
$C_{\infty,\epsilon,s}$ we have $ds_{C_{\infty,\epsilon,s}}(z)=dz/z$. 

\smallskip

We use the notation $\{ g\cdot 0, g\cdot \alpha \}_G$ to denote the homology class determined by
the image in the quotient $X_G=G\backslash \H = {\rm GL}_2({\mathbb Z})\backslash (\H^\pm \times \P)$ 
of the geodesic $C_{\alpha,\epsilon,s}$, for $g$ a representative of $s\in \P$. Similarly we
write $\{ \alpha, \beta \}_G$ for homology classes determined by the images in the
quotient $X_G$ of geodesics in $\H^\pm \times \P$ with endpoints $\alpha,\beta$ at cusps
in $\P^1({\mathbb Q})\times \P$.

\smallskip
\subsection{Limiting Modular Symbols}

We recall briefly the construction of limiting modular symbols from \cite{ManMar}.
We consider here some finite index subgroup $G\subset \Gamma$
of $\Gamma={\rm PGL}_2({\mathbb Z})$. We denote by $\P= \Gamma/G$ the finite coset space of this subgroup. 
We also write the quotient modular curve as $X_G=G\backslash \H = \Gamma\backslash (\H \times \P)$.

\smallskip

Recall that the classical modular symbols $\{ \alpha, \beta \}_G$, with $\alpha,\beta\in \P^1({\mathbb Q})$
are defined as the homology classes in $H_1(X_G,{\mathbb R})$ defined as functionals that integrate
lifts to $\H$ of differentials on $X_G$ along the geodesic arc in $\H$ connecting $\alpha$ and $\beta$, see
\cite{Man-modsymb}.
They satisfy additivity and invariance: for all $\alpha,\beta,\gamma \in  \P^1({\mathbb Q})$ 
$$ \{ \alpha,\beta \}_G + \{ \beta,\gamma \}_G =\{ \alpha, \gamma\}_G \ \ \text{ and } \ \ 
 \{ g\alpha, g\beta \}_G =\{ \alpha, \beta \}_G \ \  \forall g\in G. $$
Thus, it suffices to consider the modular symbols of the form $\{ 0, \alpha \}_G$ with $\alpha\in {\mathbb Q}$. 
These satisfy the relation
 $$ \{ 0, \alpha \}_G = - \sum_{k=1}^n  \{ \frac{p_{k-1}(\alpha)}{q_{k-1}(\alpha)},  \frac{p_k(\alpha)}{q_k(\alpha)} \}_G =
- \sum_{k=1}^n  \{ g_k^{-1}(\alpha)\cdot 0, g_k^{-1}(\alpha)\cdot \infty \}_G , $$
 where $p_k(\alpha), q_k(\alpha)$ are the successive numerators and denominators of the 
 ${\rm GL}_2({\mathbb Z})$-continued fraction expansion of $\alpha\in {\mathbb Q}$ with $p_n(\alpha)/q_n(\alpha)=\alpha$ and
 $$ g_k(\alpha)=\begin{pmatrix} p_{k-1}(\alpha) & p_k(\alpha) \\ q_{k-1}(\alpha) & q_k(\alpha) \end{pmatrix} . $$ 
 (There is an analogous formula for the ${\rm SL}_2({\mathbb Z})$-continued fraction.)
 
 \smallskip
 
 Limiting modular symbols were introduced in \cite{ManMar}, to account for the noncommutative
 compactification of the modular curves $X_G$ by the boundary $\P^1({\mathbb R})$ with the $G$ action. 
 One considers the infinite geodesics $L_\theta=\{ \infty, \theta \}$ given by the vertical lines
 $L_\theta =\{ z\in \H \,|\, \Re(z)=\theta \}$ oriented from the point at infinity to the point $\theta$ on
 the real line. Upon choosing an arbitrary base point $x\in L_\beta$ let $x(s)$ denote the point on
 $L_\beta$ at an arc-length distance $s$ from $x$ in the orientation direction. One considers the
 homology class $\{ x, x(s) \}_G \in H_1(X_G,{\mathbb R})$ determined by the geodesic arc between $x$ and $x(s)$.
 The limiting modular symbol is defined as the limit (when it exists) 
 \begin{equation}\label{limmodsymb}
 \{\{ \star, \theta \}\}_G := \lim_{s\to \infty} \frac{1}{s} \{ x, x(s) \}_G \in H_1(X_G,{\mathbb R}). 
 \end{equation}
 
 \smallskip
 
 It was shown in \cite{ManMar} that the limit \eqref{limmodsymb} exists on a full measure set 
 and can be computed by an ergodic average. More generally, it was shown in \cite{Mar} that
 there is a multifractal decomposition of the real line by level sets of the Lyapunov exponent
 of the shift of the continued fraction expansion plus an exceptional set where the limit does
 not exist. On the level sets of the Lyapunov exponent the limit is again given by an average
 of modular symbols associated to the successive terms of the continued fraction expansion.
 More precisely, as in the previous section, let $T:[0,1] \to [0,1]$ denote the shift map of the 
 continued fraction expansion,
 $$ T(x)=\frac{1}{x} - \left[ \frac{1}{x} \right], $$
 extended to a map $T: [0,1]\times \P \to [0,1]\times \P$ with $\P=\Gamma/G$ as in \eqref{Tmap}.
 The Lyapunov exponent of the shift map is given by the limit (when it exists)
 \begin{equation}\label{Lyapexp}
  \lambda(x) = \lim_{n\to \infty} \frac{1}{n} \log |(T^n)^\prime(x)| = 2\lim_{n\to \infty} \frac{1}{n} \log q_n(x) , 
 \end{equation} 
 where $q_n(x)$ are the successive denominators of the continued fraction expansion of $x\in [0,1]$. There
 is a decomposition $[0,1]=\cup_\lambda {\mathcal L}_\lambda \cup {\mathcal L}'$ where 
 ${\mathcal L}_\lambda=\{ x\in [0,1]\,|\, \lambda(x)=\lambda \}$ and ${\mathcal L}'$ the set on which the limit \eqref{Lyapexp}
 does not exist. For all $\theta\in {\mathcal L}_\lambda$ the limiting modular symbol \eqref{limmodsymb} is then given by
 \begin{equation}\label{lms2}
 \{\{ \star, \theta \}\}_G =\lim_{n\to \infty} \frac{1}{\lambda n} \sum_{k=1}^n \{ g_k^{-1}(\theta)\cdot 0, g_k^{-1}(\theta) \cdot \infty \}_G = \lim_{n\to \infty} \frac{1}{\lambda n} \sum_{k=1}^n \{ \frac{p_{k-1}(\theta)}{q_{k-1}(\theta)}, \frac{p_k(\theta)}{q_k(\theta)} \}_G .
 \end{equation}
 
 \smallskip
 
 The results of \cite{ManMar} and \cite{Mar} show that the limiting modular symbol \eqref{lms2}
 vanishes almost everywhere, with respect to the Hausdorff measure of ${\mathcal L}_\lambda$. 
 However, non-vanishing values of the limiting modular symbols are obtained, for example, 
 for all the quadratic irrationalities. 
 
 \smallskip
 
 In the case of quadratic irrationalities, one obtains two equivalent descriptions of the limiting
 modular symbol, one that corresponds to integration on the closed geodesic $C_\theta =\Gamma_\theta\backslash S_\theta$ with $S_\theta$ the infinite geodesic with endpoints the Galois conjugate pair $\theta,\theta'$ and
 the other in terms of averaged integration on the modular symbols associated to the (eventually periodic) 
 continued fraction expansion. We obtain the identification of homology classes in $H_1(X_G,{\mathbb R})$
 \begin{equation}\label{lmsRM}
  \{\{ \star, \theta \}\}_G = \frac{\sum_{k=1}^\ell \{ \frac{p_{k-1}(\theta)}{q_{k-1}(\theta)}, \frac{p_k(\theta)}{q_k(\theta)} \}_G}{\lambda(\theta) \ell } = \frac{\{ 0, g\cdot 0 \}_G}{\ell(g)}  \, \in H_1(X_G,{\mathbb R}).
 \end{equation}
 In the first expression $\ell$ is the minimal positive integer for which $T^\ell (\theta) =\theta$
 and the limit defining the Lyapunov exponent $\lambda(\theta)$ exists for quadratic irrationalities. 
 In the second expression $g\in \Gamma$ is the hyperbolic generator of $\Gamma_\theta$ with
 fixed points $\theta,\theta'$, with eigenvalues $\Lambda_g^\pm$ and 
 $\{ 0, g\cdot 0 \}_G$ denotes the homology class in $H_1(X_G,{\mathbb R})$ of the closed geodesic $C_\theta$
 and $\ell(g)=\log \Lambda_g^-=2\log \epsilon$ is the length of $C_\theta$. 
 
 \smallskip
 
 A more complete analysis of the values of the limiting modular symbols was then carried out in
 \cite{KesStra}, where it was shown that, in fact, the limiting modular symbols is non-vanishing
 on a multifractal stratification of Cantor sets of positive Hausdorff dimension. We will recall more
 precisely this result in \S \ref{BoundValSec} below. 
 
 \smallskip
 
 The construction recalled above of limiting modular symbols determine non-trivial real homology
classes in the quotient $X_G$ associated to geodesics in $\H$ with endpoints in one of the
multifractal level sets of \cite{KesStra}.  
These homology classes pair with 1-forms on $X_G$, and
in particular with weight $2$ cusp forms for the finite index subgroup $G$.

\smallskip

Let ${\mathcal M}_{G,k}$ the ${\mathbb C}$-vector space of modular forms of weight $k$ for the finite index subgroup $G\subset
{\rm GL}_2({\mathbb Z})$ and let ${\mathcal S}_{G,k}$ be the subspace of cusp forms. Let $X_G={\rm GL}_2({\mathbb Z})\backslash (\H^\pm \times \P)$
be the associated modular curve. We denote by 
\begin{equation}\label{pairSMod}
\langle\cdot, \cdot \rangle: {\mathcal S}_{G,2} \times H_1(X_G,{\mathbb R}) \to {\mathbb C}
\end{equation}
the perfect pairing between cusp forms of weight two and modular symbols., which we equivalently write as 
integration
\begin{equation}\label{pairSMod2}
 \langle \psi, \{ \alpha, \beta \}_G \rangle = \int_{\{\alpha, \beta\}_G} \psi(z)\, dz. 
\end{equation} 

\medskip
\subsection{Boundary values}\label{BoundValSec}

We consider now a linear map, constructed using cusp forms and limiting modular symbols,
that assigns to an observable of the bulk ${\rm GL}_2$-system a boundary value.

\smallskip

Let ${\mathcal L}\subset [0,1]$ denote the subset of points such that the Lyapunov exponent \eqref{Lyapexp}
of the shift of the continued fraction exists. The set ${\mathcal L}$ is stratified by level sets ${\mathcal L}_\lambda=\{
x\in [0,1]\,|\, \lambda(x)=\lambda \}$, with the Lyapunov spectrum given by the Hausdorff dimension 
function $\delta(\lambda)=\dim_H ({\mathcal L}_\lambda)$. Recall also that, for a continuous function $\phi$
on a $T$-invariant subset $E \subset [0,1]$, the Birkhoff spectrum (see \cite{FanFeng}) is the function
\begin{equation}\label{BirSpec}
 f_E(\alpha):= \dim_H {\mathcal L}_{\phi,E,\alpha},
\end{equation} 
where ${\mathcal L}_{\phi,E,\alpha}$ are the level sets of the Birkhoff average by 
\begin{equation}\label{levelBirkhoff}
 {\mathcal L}_{\phi,E,\alpha}:= \{ x \in E \,|\, \lim_{n\to \infty} \frac{1}{n}\sum_{k=0}^{n-1} \phi \circ T^k(x) =\alpha \}. 
\end{equation}
In particular, ${\mathcal L}_\lambda={\mathcal L}_{\phi,\lambda}$ for $\phi(x)=\log |T^\prime(x)|$.
Lyapunov and Birkhoff spectra for the shift of the continued fraction expansion
are analyzed in \cite{PoWe}, \cite{FLWW}.

\smallskip

In a similar way, one can consider the multifractal spectrum associated to the level
sets of the limiting modular symbol, as analyzed in \cite{KesStra}. Let $f_1,\ldots, f_g$
be a basis of the complex vectors space ${\mathcal S}_{G,2}$ of cusp forms of weight $2$ for
the finite index subgroup $G\subset {\rm GL}_2({\mathbb Z})$. Let $\Re(f_i)$, $\Im(f_i)$ be the
corresponding basis as a real $2g$-dimensional vector space. Under the pairing
\eqref{pairSMod}, \eqref{pairSMod2}, which identifies ${\mathcal S}_{G,2}$ with the dual of
$H_1(X_G,{\mathbb R})$, we can define as in \cite{KesStra} the level sets $E_\alpha$, 
for $\alpha\in {\mathbb R}^{2g}$, of the limiting modular symbol as 
\begin{equation}\label{EalphaLimMod}
E_\alpha:= \{ (x,s)\in [0,1]\times \P \,|\, ( \langle f_i, \{\{ \star, x \}\}_G \rangle =\alpha \in {\mathbb R}^{2g} \}. 
\end{equation}
Equivalently, we write this as
$$ E_\alpha =\{ (x,s) \in [0,1]\times \P \,|\, (\lim_{n\to \infty} \frac{1}{\lambda(x) n} \int_{ \{ g_k^{-1}(x)\cdot 0, g_k^{-1}(x)\cdot \infty \}_G } f_i(z)\, dz )_{i=1\ldots,g} =\alpha \in {\mathbb R}^{2g} \}. $$
For $(x,s)\in E_\alpha$ we have
$$ \lim_{n\to \infty} \frac{1}{\lambda(x) n}  \{ g_k^{-1}(x)\cdot 0, g_k^{-1}(x)\cdot \infty \}_G = h_\alpha \in H_1(X_G,{\mathbb R}), $$
where the homology class $h_\alpha$ is uniquely determined by the property that
$\langle f_i, h_\alpha\rangle =\int_{h_\alpha} f_i(z) dz = \alpha_i$. 

\smallskip

The main result of \cite{KesStra} shows that for a given finite index subgroup $G\subset {\rm GL}_2({\mathbb Z})$
with $X_G$ of genus $g\geq 1$, there is a strictly convex and differentiable function $\beta_G: {\mathbb R}^{2g}\to {\mathbb R}$
such that, for all $\alpha\in \nabla \beta_G({\mathbb R}^{2g})\subset {\mathbb R}^{2g}$
\begin{equation}\label{KSresult1}
\dim_H(E_\alpha) =\hat \beta_G(\alpha) ,
\end{equation}
where $\hat \beta_G(\alpha) =\inf_{v\in {\mathbb R}^{2g}} (\beta_G(v) - \langle \alpha, v \rangle)$ is the
Legendre transform, 
and for all $(x,s) \in E_\alpha$
\begin{equation}\label{KSresult2}
 \lim_{n\to \infty} \frac{1}{\lambda(x) n}  \{ g_k^{-1}(x)\cdot g \cdot 0, g_k^{-1}(x)\cdot g\cdot \infty \}_G = h_\alpha(x,s) 
\end{equation}
with $g$ a representative of the class $s\in \P={\rm GL}_2({\mathbb Z})/G$.

\smallskip

We let ${\mathcal A}_{\partial,G,{\mathcal P},h_\alpha}=
B_{\partial, E_\alpha}\rtimes {\rm Red}$ be the subalgebra of ${\mathcal A}_{\partial,G,{\mathcal P}}=B_\partial
\rtimes {\rm Red}$ associated to the invariant set $E_\alpha$, 
as in Corollary~\ref{BEsubalg}.
As an immediate consequence of the results \eqref{KSresult1}, \eqref{KSresult2} 
of \cite{KesStra} we have the following.

\begin{lem}\label{SG2linT}
The choice of a cusp form $\psi\in {\mathcal S}_{G,2}$ determines a bounded linear operator ${\mathcal I}_{\psi,\alpha}$ from
${\mathcal A}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$ to $B_{\partial, E_\alpha}$ with for $(x,s)\in E_\alpha$
\begin{equation}\label{Iop}
\begin{array}{rl}
\displaystyle{ {\mathcal I}_{\psi,\alpha}(f)(g,\rho,x,s)} =&\displaystyle{ \int_{ \{ \star, x \}_G } f(g,\rho,z,s) \, \psi(z)\, dz} \\[3mm] 
=& \displaystyle{
    \lim_{n\to \infty} \frac{1}{\lambda(x)\, n}\sum_{k=1}^n \int_{ \{ g_k^{-1}(x)\cdot 0, g_k^{-1}(x)\cdot \infty \}_G }  f(g,\rho,z,s) \, \psi(z)\, dz} \\[3mm]
= & \displaystyle{ \int_{h_\alpha(x)} f(g,\rho,z,s) \, \psi(z)\, dz . }    
     \end{array}
\end{equation}
\end{lem}

Here we pair the form $\omega(z)= f(z)\psi(z)dz$ with the limiting modular symbol $h_\alpha(x,s)$
of \eqref{KSresult2}. We use the notation $\omega(z)=\omega_{\rho,s}(z)$
to highlight the dependence on the variables $(\rho,s)$ that come from choosing an element $f$ in the arithmetic
algebra ${\mathcal A}^{ar}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$. 

\smallskip

Note that ${\mathcal I}_{\psi,\alpha}$ is only a linear operator and {\em not} an algebra homomorphism. 
We obtain a subalgebra of ${\mathcal A}_{\partial,G,{\mathcal P}}$ as follows.

\smallskip

\begin{defn}\label{Ialg}
Let ${\mathcal A}_{{\mathcal I},G,{\mathcal P}}$ denote the subalgebra of ${\mathcal A}_{\partial,G,{\mathcal P}}=B_\partial
\rtimes {\rm Red}$ generated by all the images ${\mathcal I}_{\psi,\alpha}(f)$ for $f\in {\mathcal A}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$,
for $\psi\in {\mathcal S}_{G,2}$, and for $\alpha \in \nabla \beta_G({\mathbb R}^{2g})$, and by the $S_k, S_k^*$
with the relations as in Definition~\ref{semicrossdef}. The arithmetic
algebra ${\mathcal A}^{ar}_{{\mathcal I},G,{\mathcal P}}$ is obtained in the same way as the algebra generated by
the images ${\mathcal I}_{\psi,\alpha}(f)$ with $f$ in the arithmetic algebra ${\mathcal A}^{ar}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$
of \S \ref{arithmSec},
for all $\psi\in {\mathcal S}_{G,2}$ and $\alpha \in \nabla \beta_G({\mathbb R}^{2g})$, and by the $S_k,S_k^*$ as
above.
\end{defn}

\medskip
\subsection{Evaluation of ground states on boundary values}

When we evaluate zero-temperature KMS states on the elements ${\mathcal I}_{\psi,\alpha}(f)$, for
an element $f\in {\mathcal A}^{ar}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$, we
obtain the pairing of a cusp form with a limiting modular symbol,
\begin{equation}\label{KMSIpsi}
 \varphi_{\infty,\rho,x,s}({\mathcal I}_{\psi,\alpha}(f)) =\langle  \omega_{\rho,s}, h_\alpha(x) \rangle, 
\end{equation} 
where $\omega_{\rho,s}(z)=f(1,\rho,z,s) \psi(z) dz$ is a cusp form in ${\mathcal S}_{G,2}$ for all $(\rho,s)$.  
Since elements $f\in {\mathcal A}^{ar}_{{\rm GL}_2({\mathbb Z}),G,{\mathcal P}}$
depend on the variable $\rho\in M_2(\hat{\mathbb Z})$ through some finite projection $\pi_N(\rho)\in {\mathbb Z}/N{\mathbb Z}$,
we can write $\omega_{\rho,s}(z)$ as a finite collection $\{ \omega_{i,s}(z) \}_{i\in {\mathbb Z}/N{\mathbb Z}}$. 

\smallskip

To illustrate the properties of the values of ground states on arithmetic
boundary elements, we consider here the particular case where $G=\Gamma_0(N)$
and a state $\varphi_{\infty,\rho,x,s}$ with $s\in \P$. 
We also choose $f$ and $\psi$ so that the resulting $\omega_{i,s}$ are 
cusp forms for $\Gamma_0(N)$ that are Hecke eigenforms for all the Hecke
operators $T(m)$.

\smallskip

Recall (see \cite{Serre}) that the Hecke operators $T(m)$ given by 
$$ T_m=\sum_{\gamma\,:\, \det(\gamma)=m} \Gamma_0(N) \gamma \Gamma_0(N) $$
satisfying $T_nT_m=T_mT_n$ for $(m,n)=1$ and $T_{p^n} T_p=T_{p^{n+1}} + p T_{p^{n-1}} R_p$
with $R_\lambda$ the scaling operator that acts on a modular form of weight $2k$ as
multiplication by $\lambda^{-2k}$. The Hecke operators $T_m$ and the scaling operators $R_\lambda$ 
generate a commutative algebra, and the action of $T_m$ on a modular form of weight $2k$ is
given by
$$ T_m\, f(z) = n^{2k-1} \sum_{a\geq 1, ad=n, 0\leq b <d} d^{-2k} \, f(\frac{az+b}{d}). $$

\smallskip

\begin{prop}\label{evalKMSinftyMerel}
Let $G=\Gamma_0(N)$ and let $\omega_{i,s_0}$, with $i=0,\ldots, N-1$ and $s_0=\Gamma_0(N) g_0 \in \P$, 
be Hecke eigencuspforms of weight $2$. For $s\in \P$ with $s=\Gamma_0(N) g_0 \gamma$, with 
$\gamma\in {\rm GL}_2({\mathbb Z})$, let $\omega_{i,s}=\omega_{i,s_0 \gamma}:=\omega_{i,s_0}\circ \gamma^{-1}$. 
Consider the pairing 
$$ \xi_\omega(s)= \langle  \omega_{\rho,s}, h_\alpha(x) \rangle =\int_{h_\alpha(x)} \omega_{i,s} $$
with the limiting modular symbol $h_\alpha(x)$, as in \eqref{Iop} and \eqref{KMSIpsi}. For $(m,N)=1$
we have the relations
$$ a_{i,m} \,  \xi_\omega(s) = \sum_{M \in \bar A_m} u_m\, \xi_\omega(s\, M) $$
where $a_{i,m}$ are the Hecke eigenvalues and 
$A_m=\{ M \in M_2({\mathbb Z})\,:\, \det(M)=m \}$, with $\bar A_m =A_m/\{ \pm 1\}$ and
$\sum_M u_M\, M \in {\mathbb Z} \bar A_m$ is the Manin--Heilbronn lift of the Hecke operator $T_m$.
\end{prop}


\proof The condition that $\omega_{i,s \gamma}=\omega_{i,s}\circ \gamma^{-1}$ implies
that 
$\langle  \omega_{i,s}, h_\alpha(x) \rangle=\langle \omega_{i,s_0}, h_\alpha(x,s) \rangle$.
The following facts are known from \cite{Man-modsymb}, \cite{Merel}.  Let 
$$ A_{m,N}=\{ M=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in M_2({\mathbb Z})\,:\, \det(M)=m, \, \, N|c  \}. $$
Let $R$ be a set of representatives for the classes $\Gamma_0(N)\backslash A_{m,N}$
The Hecke operators act on the modular symbols by 
$T_m \{ \alpha, \beta \}=\sum_{\lambda \in R} \{ \lambda \alpha, \lambda \beta \}$. 
For $(m,N)=1$ there is a bijection between the cosets $\Gamma_0(N)\backslash A_{m,N}$
and $A_m/{\rm SL}_2({\mathbb Z})$. For $s\in \P$ consider the assignment $\xi_\omega: s\mapsto \xi_\omega(s)= \langle \omega_{i,s}, h_\alpha(x,s)\rangle$,
where $\omega_{i,s}$ is a Hecke eigencuspform. It is shown in  \cite{Man-modsymb}, \cite{Merel}
that there is a lift $\Theta_m$ of the action of the Hecke operators $T_m \circ \xi=\xi\circ \Theta_m$ 
(the Manin--Heilbronn lift), which is given by $\Theta_m =\sum_{\gamma \in A_m /{\rm SL}_2({\mathbb Z})} \Upsilon_\gamma$,
where $\Upsilon_\gamma$ is a formal chain of level $m$ connecting $\infty$ to $0$ and of class $\gamma$.
This means that $\Upsilon_\gamma=\sum_{k=0}^{n-1} \gamma_k$ in ${\mathbb Z} A_m$, for some $n\in {\mathbb N}$ where
$$ \gamma_k =\begin{pmatrix} u_k & u_{k+1} \\ v_k & v_{k+1} \end{pmatrix} $$
with $u_0/v_0=\infty$ and $u_n/v_n=0$ and where $\gamma_k$ agrees with $\gamma$ 
in $A_m/{\rm SL}_2({\mathbb Z})$. An argument in \cite{Merel} based on the continued fraction expansion 
and modular symbols shows that it is always possible to construct such formal chains with $\gamma_k =\gamma g_k$ with $g_k\in {\rm SL}_2({\mathbb Z})$ and
that the resulting $\Theta_m$ is indeed a lift of the Hecke operators. (The length $n$ of the chain
of the Manin--Heilbronn lift is also computed, see \S 3.2 of \cite{Merel}.) Using the notation of
Theorem~4 of \cite{Merel}, we write the Manin--Heilbronn lift as 
$\Theta_m = \sum_{M\in \bar A_m} u_M\, M$ as an element of ${\mathbb Z} A_m$. Each element 
$M\in \bar A_m$ maps $s\mapsto s\, M$ in ${\mathcal P}$, hence one obtains a map $\Theta_m: {\mathbb Z}{\mathcal P} \to {\mathbb Z}{\mathcal P}$. 
In particular, as shown in Theorem~2 of \cite{Merel}, for $s=\Gamma_0(N)g$ in $\P$ one has
$\Theta_m(s)=\sum_{\gamma\in R} \sum_{k=0}^{n-1} \phi(g\gamma \gamma_k)$ where 
$\phi: A_m \to \Gamma_0(N)\backslash {\rm SL}_2({\mathbb Z})$ is the map that assigns 
$$ A_m \ni \gamma=\begin{pmatrix} a & b \\ c & d \end{pmatrix} \mapsto \Gamma_0(N) 
\begin{pmatrix} w & t \\ u & v \end{pmatrix} \in \Gamma_0(N) \backslash {\rm SL}_2({\mathbb Z}) $$
with $(c :d)=(u :v)$ in $\P^1({\mathbb Z}/N{\mathbb Z})\simeq \P =\Gamma_0(N) \backslash {\rm SL}_2({\mathbb Z})$.
Thus, as in Theorem~2 of \cite{Merel} we then have 
$\Theta_m(s)=\sum_{\gamma\in R} \sum_{k=0}^{n-1} \phi(g\gamma) g_k=\sum_{\gamma\in R} \phi(g\gamma) \gamma^{-1}$, seen here as an element in ${\mathbb Z}{\mathcal P}$. Thus, in the pairing of limiting modular symbols and
boundary elements we find
$$ T_m \xi_\omega(s) =T_m \int_{h_\alpha(x,s)} \omega_{i,s_0} =\int_{h_\alpha(x,s)} T_m \omega_{i,s_0}
= a_{i,m} \int_{h_\alpha(x,s)} \omega_{i,s_0} ,$$
where $a_{i,m}$ are the Hecke eigenvalues of the eigenform $\omega_{i,s_0}$, with $a_{i,1}=1$.
On the other hand, using the Manin--Heilbronn lift we have
$$ T_m \xi_\omega(s) = \xi_\omega (\Theta_m(s)) =
\sum_{\gamma\in R} \int_{h_\alpha(x,s_\gamma)} \omega_{i,s_0} $$
with $s_\gamma \in {\mathcal P}$ given by $s_\gamma=\phi(g\gamma) \gamma^{-1}$ as above.
We write the latter expression in the form 
$$ \sum_{M\in \bar A_m} u_M\, \int_{h_\alpha(x,s\, M)} \omega_{i,s_0} $$
for consistency with the notation of Theorem~4 of \cite{Merel}. 
\endproof

\smallskip

\begin{prop}\label{L1period}
Under the same hypothesis as Proposition~\ref{evalKMSinftyMerel}, 
let $L_{\omega_{i,s_0}}(\sigma)=\sum_m a_{i,m} \, m^{-\sigma}$ be the $L$-series associated to the
cusp form $\omega_{i,s_0}=\sum_m a_{i,m} \, q^m$. For $x$ a quadratic irrationality 
the evaluation \eqref{KMSIpsi} of KMS$_\infty$ states satisfies
\begin{equation}\label{L1eval}
 \langle \omega_{i,s}, h(x) \rangle=
 \frac{1}{\lambda(x) n} \sum_{k=1}^n \langle \omega_{i,s_k}, \{0, \infty\} \rangle
= \frac{1}{\lambda(x) n} \sum_{k=1}^n L_{\omega_{i,s_k}}(1),
\end{equation}
where $s_k=\Gamma_0(N) g g_k^{-1}(x)$ for $s=\Gamma_0(N) g$.
\end{prop}

\proof
The special value $L_{\omega_{i,s_0}}(1)$ of the $L$-function gives the pairing with the
modular symbol $\langle \omega_{i,s_0}, \{ 0, \infty\} \rangle$. Similarly, for $s\in \P$
with $s=\Gamma_0(N)g$, the special value gives $$L_{\omega_{i,s}}(1)=\langle \omega_{i,s}, \{ 0, \infty\} \rangle
=\langle \omega_{i,s_0}, \{ g\cdot 0, g\cdot \infty\} \rangle.$$ 
In the case of a quadratic irrationality the limiting modular symbol satisfies
$$ h(x)=\frac{1}{\lambda(x) n} \sum_{k=1}^n \{ g_k^{-1}(x)\cdot 0, g_k^{-1}(x) \cdot \infty \}_G, $$
where $n$ is the length of the period of the continued fraction expansion of $x$ and $\lambda(x)$
is the Lyapunov exponent. For $s_k=\Gamma_0(N) g g_k^{-1}(x)$ with $s=\Gamma_0(N) g$, we have
$\langle \omega_{i,s_k}, \{0, \infty\} \rangle = \langle \omega_{i,s}, \{ g_k^{-1}(x)\cdot 0, g_k^{-1}(x) \cdot \infty \}\rangle$ hence one obtains \eqref{L1eval}.
\endproof

\smallskip

 As shown in Theorem~3.3 of
\cite{Man-modsymb}, the special value $L_{\omega_{i,s_0}}(1)$ satisfies 
$$ (\sum_{d|m} d - a_{i,m}) L_{\omega_{i,s_0}}(1) = \sum_{d|m, \, b \, {\rm mod} \, d} 
\int_{\{ 0, b/d \}_G} \omega_{i,s_0}, $$
since one has
$$ \int_0^{\infty} T_m \omega_{i,s_0} = a_{i,m} \int_0^{\infty} \omega_{i,s_0} =
\sum_{d|m} \sum_{b=0}^{d-1} \int_{\{ b/d,0 \}_G +\{ 0, \infty \}_G} \omega_{i,s_0}. $$

\smallskip

For a normalized Hecke eigencuspform $f=\sum_n a_n q^n$, let 
$L_f(s)=\sum_n a_n n^{-s}$ be the associated $L$-function and
$\Lambda_f(s)=(2\pi)^{-s}\Gamma(s) L_f(s)$ the completed $L$-function,
the Mellin transform $\Lambda_f(s)=\int_0^\infty f(iz) z^{s-1} dz$.

\smallskip

The relation between special values of $L$-functions and periods of Hecke
eigenforms generalizes for higher weights, and it was shown in \cite{Man-Hecke}
that ratios of these special values of the same parity are algebraic (in the field
generated over ${\mathbb Q}$ by the Hecke eigenvalues).
For a normalized Hecke eigencuspform $f=\sum_n a_n q^n$ of weight $k$ 
the coefficients of the period polynomial $r_f(z)$ are expressible in terms of special
values of the $L$-function,
$$ r_f(z)=-i \sum_{j=0}^{k-2} \binom{k-2}{j} (iz)^j \Lambda_f(j+1). $$
Manin's Periods Theorem shows that, for ${\mathbb K}_f$ the field of algebraic numbers 
generated over ${\mathbb Q}$ by the Fourier coefficients, there are $\omega_\pm(f)\in {\mathbb R}$
such that for all $1\leq s \leq k-1$ with $s$ even $\Lambda_f(s)/\omega_+(f)\in {\mathbb K}_f$,
respectively $\Lambda_f(s)/\omega_-(f)\in {\mathbb K}_f$ for $s$ odd.

\smallskip

Shokurov gave a geometric argument based on Kuga varieties and a higher
weight generalization of modular symbols, \cite{Shok}.  It is expected that the
limiting modular symbols of \cite{ManMar}, as well as the quantum statistical mechanics 
of the ${\rm GL}_2$-system and its boundary described here, will generalize to the case of
Kuga varieties, with the relations between periods of Hecke eigencuspforms described
in \cite{Man-Hecke} arising in the evaluation of zero-temperature KMS states of these
systems. 


\bigskip

\subsection*{Acknowledgment} The first author is partially supported by 
NSF grant DMS-1707882, by NSERC Discovery Grant RGPIN-2018-04937 
and Accelerator Supplement grant RGPAS-2018-522593, and the
Perimeter Institute for Theoretical Physics. 

