Toric $AF$-algebras and faithful representations of the mapping class group
Igor Nikolaev \footnote{Partially supported by NSERC.}
We construct a covariant non-injective functor from the space of generic Riemann surfaces to the so-called toric $AF$-algebras; such a functor maps isomorphic Riemann surfaces into stably isomorphic toric $AF$-algebras. We use the functor to construct a faithful representation of the mapping class group of genus $g>1$ into the matrix group $GL_{6g-6}({\Bbb Z})$. \vspace{7mm} {Key words and phrases: Riemann surfaces, $AF$-algebras} \vspace{5mm} {AMS (MOS) Subj. Class.: 14H55, 20F65, 46L85}
math0111173.tex
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\title{Toric $AF$-algebras  and faithful representations
of the mapping class group}


\author{Igor Nikolaev
\footnote{Partially supported 
by NSERC.}}

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\date{}
 \maketitle
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\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}
\newtheorem{dfn}{Definition}
\newtheorem{rmk}{Remark}
\newtheorem{cor}{Corollary}
\newtheorem{prp}{Proposition}
\newtheorem{exm}{Example}
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\begin{abstract}
We  construct a covariant non-injective functor
from the space of generic Riemann surfaces to the so-called
toric $AF$-algebras;  such a functor maps isomorphic  Riemann surfaces 
into  stably isomorphic toric  $AF$-algebras.  We use the functor  to 
construct a faithful representation of the mapping class group of genus 
$g>1$  into the matrix group $GL_{6g-6}({\Bbb Z})$. 




\vspace{7mm}

{\it Key words and phrases:  Riemann surfaces, $AF$-algebras}

\vspace{5mm}
{\it AMS (MOS) Subj. Class.:  14H55, 20F65, 46L85}
\end{abstract}


%**************************************************************************
\section{Introduction}
%***************************************************************************
{\bf A. The mapping class group.}
The mapping class group has  been introduced in the 1920-ies by M.~Dehn \cite{Deh1}. 
Such a  group, $Mod~(X)$, is defined as the group of isotopy classes of the
orientation-preserving diffeomorphisms of a two-sided closed surface $X$ of
genus $g\ge 1$.  The group is known to be  prominent in  algebraic geometry \cite{HaLo1}, 
topology \cite{Thu1}  and dynamics \cite{Thu2}.  When $X$
is a   torus, the $Mod~(X)$ is isomorphic to the group  $SL_2({\Bbb Z})$. 
(The $SL_2({\Bbb Z})$ is called a modular group, hence our notation for the mapping class groups.) 
A little is known about the representations of $Mod~(X)$ beyond the case $g=1$.
Recall, that the group is called {\it linear},  if there exists a faithful
representation into the matrix group $GL_m(R)$, where $R$ is a commutative ring.
The braid groups are known  to be linear \cite{Big1}.  Using a modification of the argument for 
the braid groups, it is possible to prove, that $Mod~(X)$ is linear in the case $g=2$
\cite{BiBu1}. Whether the mapping class group is linear for  $g\ge 3$, 
is an open problem, known as a {\it Harvey conjecture} \cite{Har1}, p.267. 



\medskip\noindent
{\bf B. The toric $AF$-algebras.}
 Denote by $T_S(g)$ the Teichm\"uller
space of genus $g\ge 1$ with a distinguished point $S$. Let
$q\in H^0(S, \Omega^{\otimes 2})$ be a holomorphic quadratic 
differential on the Riemann surface $S$, such that all zeroes
of $q$ (if any) are simple. By $\widetilde S$ we mean a double 
cover of $S$ ramified over the zeroes of $q$ and by
$H_1^{odd}(\widetilde S)$ the odd part of the integral homology of $\widetilde S$ relative to the  zeroes.
Note that $H_1^{odd}(\widetilde S)\cong {\Bbb Z}^{n}$, where $n=6g-6$ if $g\ge 2$ and $n=2$ if $g=1$. 
The fundamental result of Hubbard and Masur \cite{HuMa1} implies, that
$T_S(g)\cong Hom~(H_1^{odd}(\widetilde S); {\Bbb R})-\{0\}$, where $0$ is the zero homomorphism
(see \S 3.1 for the details). 
Finally, denote by $\lambda=(\lambda_1,\dots,\lambda_{n})$ the image of a basis of 
$H_1^{odd}(\widetilde S)$ in the real line ${\Bbb R}$, such that $\lambda_1\ne0$. 
Note that such an option always exists, since the zero homomorphism is excluded.
We let $\theta=(\theta_1,\dots,\theta_{n-1})$, where $\theta_i=\lambda_{i-1}/\lambda_1$. 
Recall that,  up to a scalar multiple, the vector $(1,\theta)\in {\Bbb R}^{n}$ is the limit
of a generically convergent Jacobi-Perron continued fraction \cite{B}:
%**************************************************************
$$
\left(\matrix{1\cr \theta}\right)=
\lim_{k\to\infty} \left(\matrix{0 & 1\cr I & b_1}\right)\dots
\left(\matrix{0 & 1\cr I & b_k}\right)
\left(\matrix{0\cr {\Bbb I}}\right),
$$
%****************************************************************
where $b_i=(b^{(i)}_1,\dots, b^{(i)}_{n-1})^T$ is a vector of the non-negative integers,  
$I$ the unit matrix and ${\Bbb I}=(0,\dots, 0, 1)^T$. We introduce an  $AF$-algebra, 
${\Bbb A}_{\theta}$, via the Bratteli diagram \cite{E}, shown in Fig.1.
(The numbers $b_j^{(i)}$ of the diagram indicate the multiplicity  of edges of 
the graph.)  Let us call ${\Bbb A}_{\theta}$ a {\it toric $AF$-algebra}. 
Note that in the $g=1$ case, the Jacobi-Perron 
fraction coincides with the  regular continued fraction and ${\Bbb A}_{\theta}$
becomes the Effros-Shen $AF$-algebra of a noncommutative torus
\cite{EfSh1}.
    


%\bigskip
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\put(88,120){$b_1^{(1)}$}


\put(185,5){$b_5^{(2)}$}
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\end{picture}


\caption{A toric $AF$-algebra ${\Bbb A}_{\theta}$ of genus $g=2$.}
\end{figure}
%*******************************************************************






\medskip\noindent
{\bf C. An objective.}
It is known, that  $Mod~(X)$ acts on the space $T_S(g)$ by 
isomorphisms of the Riemann surfaces \cite{HaLo1}. Let $R,R'\in T_S(g)$ be a pair of isomorphic 
Riemann surfaces;  let ${\Bbb A}_{\theta}$ and ${\Bbb A}_{\theta'}$ be the 
corresponding  toric $AF$-algebras.  Our objective can be formulated as follows.


\medskip\noindent
{\sf Main problem.} 
{\it (i) to find an equivalence relation between the toric $AF$-algebras 
${\Bbb A}_{\theta}$ and ${\Bbb A}_{\theta'}$;
(ii) to construct a  (faithful)  representation of the mapping class group induced by the 
above equivalence relation.}


\medskip\noindent
Recall,  that the stable isomorphism between the $C^*$-algebras is a
fundamental equivalence in noncommutative geometry; the $C^*$-algebras
${\Bbb A}$ and ${\Bbb A}'$ are said to be {\it stably isomorphic}, whenever ${\Bbb A}\otimes {\cal K}$
is isomorphic to  ${\Bbb A}'\otimes {\cal K}$, where ${\cal K}$ is the $C^*$-algebra
of compact operators.  It turns out,  that the  difficult  question  of  isomorphism   between the
Riemann surfaces  has an  elegant answer in terms of the $C^*$-algebras. 
Roughly, an  answer to (i) is this --  the toric $AF$-algebras ${\Bbb A}_{\theta}$ and ${\Bbb A}_{\theta'}$
must be  stably isomorphic -- see theorem \ref{thm1}.  Unfortunately, this beautiful fact holds only for the  typical (generic)
Riemann surfaces, which we will specify further.  An obstacle in the remaining cases  
is a (non-generic) divergence of the Jacobi-Perron fractions. 
Only in the $g=1$ case one always does have  convergence, i.e. when ${\Bbb A}_{\theta}$ is the Effros-Shen
 algebra \cite{EfSh1}.  The above action of $Mod~(X)$ on the toric $AF$-algebras is remarkable:  
it is known,  that whenever the automorphism group of such an algebra is trivial (this is generically true), 
the stable isomorphism group of the toric $AF$-algebra admits a faithful representation into
a matrix group over the commutative ring ${\Bbb Z}$ \cite{E}.  This elementary observation  yields an amazingly 
simple  positive solution to the Harvey conjecture (theorem \ref{thm2}).   

  



\medskip\noindent
{\bf D. The results.}
Denote by $V$ the maximal subset of $T_S(g)$,  such that for each Riemann surface $R\in V$, 
there exists a convergent Jacobi-Perron continued fraction. 
Let $F$ be the map which sends the Riemann surfaces into the toric $AF$-algebras
according to the formula  $R\mapsto {\Bbb A}_{\theta}$.  
Finally, let $W$ be the image of  $V$ under the mapping $F$.
A  summary of  our results is given below.
%****************************************************************
\begin{thm}\label{thm1}
The set $V$ is a generic subset of $T_S(g)$ and the  map $F$ has the following properties:

\medskip
(i) $V\cong W\times (0,\infty)$ is a trivial fiber bundle, whose
projection  map $p: V\to W$ coincides with $F$;


\smallskip
(ii) $F: V\to W$ is a covariant functor, which maps isomorphic Riemann surfaces 
$R,R'\in V$ to  stably isomorphic toric $AF$-algebras ${\Bbb A}_{\theta},{\Bbb A}_{\theta'}\in W$.
\end{thm}
%***************************************************************
%*******************************************************************
\begin{thm}\label{thm2}
For every surface $X$ of  genus $g\ge 2$,  there exists a faithful representation 
$\rho: Mod~(X)\rightarrow GL_{6g-6}({\Bbb Z})$.
\end{thm}
%******************************************************************* 
(Note that theorem \ref{thm1} has been proved in \cite{Nik1}; 
for the sake of clarity,  we repeat the original argument below.)
The structure of the article is as follows. 
Theorems \ref{thm1} and \ref{thm2} are proved in section 2;
section 3 is an appendix containing  the notation and results,
which are vital  for the proof of theorems \ref{thm1} and \ref{thm2}.   






%**************************************************************************
\section{Proof}
%***************************************************************************
%**************************************************************************
\subsection{Proof of theorem 1}
%***************************************************************************
Let us outline the proof. We consider the following categories:
(i) generic Riemann surfaces ${\cal V}=V\subset T(g)$;
(ii) pseudo-lattices ${\cal PL}$; (iii) projective pseudo-lattices
${\cal PPL}$ and (iv)  category ${\cal W}$ of the toric $AF$-algebras.
First, we show that ${\cal V}\cong {\cal PL}$ are equivalent categories,  so that isomorphic
Riemann surfaces $R,R'\in {\cal V}$ map to isomorphic pseudo-lattices
$PL, PL'\in {\cal PL}$.  Next, a non-injective functor $F: {\cal PL}\to {\cal PPL}$
is constructed;  $F$ maps isomorphic pseudo-lattices to isomorphic
projective pseudo-lattices and $Ker ~F\cong (0,\infty)$. 
Finally, it is shown that a subcategory $U\subseteq {\cal PPL}$ and ${\cal W}$ are the 
equivalent categories. In other words, we have the following diagram:
%**************************************************************
\begin{equation}\label{eq8}
{\cal V}
\buildrel\rm\alpha
\over\longrightarrow
{\cal PL}
\buildrel\rm F
\over\longrightarrow
U
\buildrel\rm \beta
\over\longrightarrow
{\cal W},
\end{equation}
%********************************************************************
where $\alpha$ is an injective map, $\beta$ is a bijection  and $Ker ~F\cong (0,\infty)$. 


\bigskip\noindent
\underline{Category ${\cal V}$}. 
Let $Mod ~(X)$ be the mapping class group of the surface $X$. 
A {\it Riemann surface}  is a triple $(X, R, j)$, where
$X$ is a topological surface of genus $g\ge 1$,  $j: X\to R$ is a
complex (conformal) parametrization of $X$ and $R$ is a Riemann surface.
A {\it morphism} of Riemann surfaces  
$(X, R, j)\to (X, R', j')$   is the identity
$j\circ\psi=\varphi\circ j'$, 
where $\varphi\in Mod~(X)$ is a diffeomorphism of $X$  and $\psi$ is an isomorphism of the 
Riemann surfaces. 
A  category of generic Riemann surfaces, ${\cal V}$, consists of $Ob~({\cal V})$
which are Riemann surfaces  $R\in V\subset T_S(g)$ and morphisms $H(R, R')$
between $R, R'\in Ob~({\cal V})$ which coincide with the morphisms
specified above. For any  $R, R', R''\in Ob~({\cal V})$
and any morphisms $\varphi': R\to R'$, $\varphi'': R'\to R''$ a 
morphism $\phi: R\to R''$ is the composite of $\varphi'$ and
$\varphi''$, which we write as $\phi=\varphi''\varphi'$. 
The identity  morphism, $1_R$, is a morphism $H(R,R)$.





\bigskip\noindent
\underline{Category ${\cal PL}$}. 
A {\it pseudo-lattice}
%**************************************************************************
\footnote{See \cite{Man1}.}
%**************************************************************************
 (of rank $n$) is a triple $(\Lambda, {\Bbb R}, j)$, where
$\Lambda\cong {\Bbb Z}^n$ and $j: \Lambda\to {\Bbb R}$ is a homomorphism.  
A morphism of pseudo-lattices $(\Lambda, {\Bbb R}, j)\to (\Lambda, {\Bbb R}, j')$
is the identity
$j\circ\psi=\varphi\circ j'$,
where $\varphi$ is a group homomorphism and $\psi$ is an inclusion 
map, i.e. $j'(\Lambda')\subseteq j(\Lambda)$.  
Any isomorphism class of a pseudo-lattice contains
a representative given by $j: {\Bbb Z}^n\to  {\Bbb R}$ such that 
%**********************************************************************
$j(1,0,\dots, 0)=\lambda_1, \quad j(0,1,\dots,0)=\lambda_2,\quad \dots, \quad  j(0,0,\dots,1)=\lambda_n,$
%***************************************************************************** 
where $\lambda_1,\lambda_2,\dots,\lambda_n$ are positive reals.
The pseudo-lattices of rank $n$  make up a category, which we denote by ${\cal PL}_n$. 
%**********************************************************************
\begin{lem}{\bf (basic lemma)}\label{lm5}
Let $g\ge 2$ (respectively $g=1$) and $n=6g-6$ (respectively $n=2$). 
There exists an injective covariant functor $\alpha: {\cal V}\to {\cal PL}_{n}$,
which maps isomorphic Riemann surfaces $R, R'\in {\cal V}$ to the isomorphic
pseudo-lattices $PL,PL'\in {\cal PL}_{n}$.
\end{lem}
%**********************************************************************
{\it Proof.} Let $\alpha: T(g)-\{pt\}\to Hom~(H_1^{odd}(\widetilde S); {\Bbb R})-\{0\}$
be the map  constructed in \S 1.1. By the Hubbard-Masur theorem, $\alpha$
is a homeomorphism. In particular, $\alpha$ is an injective map. 



Let us find the image $\alpha(\varphi)\in Mor~({\cal PL})$ of 
$\varphi\in Mor~({\cal V})$. Let $\varphi\in Mod~(X)$ be a diffeomorphism
of $X$, and let $\widetilde X\to X$ be the ramified double cover of $X$. 
We denote by $\widetilde\varphi$ the induced
mapping on $\widetilde X$. Note that $\widetilde\varphi$ is a diffeomorphism
of $\widetilde X$ modulo the covering involution ${\Bbb Z}_2$. Denote by
$\widetilde\varphi^*$ the action of $\widetilde\varphi$ on 
$H_1^{odd}(\widetilde X)\cong {\Bbb Z}^{n}$. Since $\widetilde\varphi~mod~{\Bbb Z}_2$ is
a diffeomorphism of $\widetilde X$, $\widetilde\varphi^*\in GL_{n}({\Bbb Z})$. 
Thus, $\alpha(\varphi)=\widetilde\varphi^*\in Mor~({\cal PL})$. 


Let us show that $\alpha$ is a functor.  Indeed, let $R, R'\in {\cal V}$ be isomorphic
Riemann surfaces, such that $R'=\varphi(R)$ for a $\varphi\in Mod~(X)$. 
Let $a_{ij}$ be the elements of matrix $\widetilde\varphi^*\in GL_{n}({\Bbb Z})$. 
Recall that $\lambda_i=\int_{\gamma_i}\phi$ for a closed 1-form $\phi= Re~\omega$
and $\gamma_i\in H_1^{odd}(\widetilde X)$. Then 
$\gamma_j=\sum_{i=1}^{n} a_{ij}\gamma_i, \quad j=1,\dots, n$
are the elements of a new basis in $H_1^{odd}(\widetilde X)$. By the integration rules,
%*******************************************************************
\begin{equation}\label{eq10}
\lambda_j'= \int_{\gamma_j}\phi=
\int_{\sum a_{ij}\gamma_i}\phi=
\sum_{i=1}^{n}a_{ij}\lambda_i.
\end{equation}
%****************************************************************
Finally, let $j(\Lambda)={\Bbb Z}\lambda_1+\dots+{\Bbb Z}\lambda_{n}$
and $j'(\Lambda)={\Bbb Z}\lambda_1'+\dots+{\Bbb Z}\lambda_{n}'$.
Since $\lambda_j'= \sum_{i=1}^{n}a_{ij}\lambda_i$ and $(a_{ij})\in GL_{n}({\Bbb Z})$,
we conclude that $j(\Lambda)=j'(\Lambda)\subset {\Bbb R}$. In other words, the pseudo-lattices
$(\Lambda, {\Bbb R}, j)$ and $(\Lambda, {\Bbb R}, j')$ are isomorphic. Hence, 
$\alpha: {\cal V}\to {\cal PL}$  maps isomorphic Riemann surfaces to
the isomorphic pseudo-lattices, i.e. $\alpha$ is a functor.  


Finally, let us show that $\alpha$ is a covariant functor. 
Indeed, let $\varphi_1,\varphi_2\in Mor ({\cal V})$. Then  
$\alpha(\varphi_1\varphi_2)= (\widetilde{\varphi_1\varphi_2})^*=\widetilde\varphi_1^*\widetilde\varphi_2^*=
\alpha(\varphi_1)\alpha(\varphi_2)$. Lemma \ref{lm5} follows.
$\square$






\bigskip\noindent
\underline{Category ${\cal PPL}$}. 
A  {\it projective pseudo-lattice} (of rank $n$) is a triple 
$(\Lambda, {\Bbb R}, j)$, where $\Lambda\cong {\Bbb Z}^n$ and $j: \Lambda\to {\Bbb R}$ is a
homomorphism. A morphism of projective pseudo-lattices
$(\Lambda, {\Bbb R}, j)\to (\Lambda, {\Bbb R}, j')$
is the identity
$j\circ\psi=\varphi\circ j'$,
where $\varphi$ is a group homomorphism and $\psi$ is an  ${\Bbb R}$-linear
map. (Notice that unlike the case of pseudo-lattices, $\psi$ is a scaling
map as opposed to an inclusion map. This allows the two pseudo-lattices
to be projectively equivalent, while being distinct in the category ${\cal PL}_n$.) 
It is not hard to see that any isomorphism class of a projective pseudo-lattice 
contains a representative given by $j: {\Bbb Z}^n\to  {\Bbb R}$ such that
%***************************************************************************
$j(1,0,\dots,0)=1,\quad
j(0,1,\dots,0)=\theta_1,\quad  \dots, \quad  j(0,0,\dots,1)=\theta_{n-1},$
%***************************************************************************
  where $\theta_i$ are  positive reals.
The projective pseudo-lattices of rank $n$  make up a category, which we denote by ${\cal PPL}_n$. 





\bigskip\noindent
\underline{Category $W$}. 
Finally, the toric $AF$-algebras ${\Bbb A}_{\theta}$,
modulo the stable isomorphism between them,  make up a category, 
which we shall denote by $W_n$. 
%***************************************************************
\begin{lem}\label{lm6}
Let $U_n\subseteq {\cal PPL}_n$ be a subcategory consisting of the projective
pseudo-lattices $PPL=PPL(1,\theta_1,\dots,\theta_{n-1})$ for which the Jacobi-Perron
fraction of the vector $(1,\theta_1,\dots,\theta_{n-1})$ converges to the vector.
Define a map  $\beta: U_n\to W_n$  by the formula
%********************************************************************
$PPL(1,\theta_1,\dots,\theta_{n-1})\mapsto {\Bbb A}_{\theta},$
%*******************************************************************
where $\theta=(\theta_1,\dots,\theta_{n-1})$. 
Then $\beta$ is a bijective functor, which maps isomorphic projective pseudo-lattices 
to the stably isomorphic  toric $AF$-algebras. 
 \end{lem}
%***************************************************************
{\it Proof.} It is evident that $\beta$ is injective and surjective. Let
us show that $\beta$ is a functor. Indeed,  according to 
 \cite{E}, Corollary 4.7, every totally 
ordered abelian group of rank $n$ has form ${\Bbb Z}+\theta_1 {\Bbb Z}+\dots+ {\Bbb Z}\theta_{n-1}$.
The latter   is a projective pseudo-lattice $PPL$  from the category $U_n$. 
On the other hand, by  the Elliott theorem \cite{Ell1}, the $PPL$ defines a stable isomorphism class of 
the toric $AF$-algebra ${\Bbb A}_{\theta}\in W_n$. 
Therefore, $\beta$  maps isomorphic projective pseudo-lattices (from the set $U_n$) to the stably isomorphic toric 
$AF$-algebras,  and {\it vice versa}. Lemma \ref{lm6} follows.  
 $\square$



\medskip
Let $PL(\lambda_1,\lambda_2,\dots, \lambda_n)\in {\cal PL}_n$ and 
$PPL(1,\theta_1,\dots,\theta_{n-1})\in {\cal PPL}_n$. 
To finish the proof of theorem \ref{thm1}, it remains to prove the following lemma.
%***************************************************************
\begin{lem}\label{lm7}
Let $F: {\cal PL}_n\to {\cal PPL}_n$ be a map given by
$PL(\lambda_1,\lambda_2,\dots, \lambda_n)\mapsto PPL\left(1, {\lambda_2\over\lambda_1},\dots, {\lambda_n\over\lambda_1}\right).$
Then $Ker~F=(0,\infty)$ and  $F$ is a functor which maps  isomorphic pseudo-lattices to isomorphic projective 
pseudo-lattices. 
\end{lem}
%*************************************************************** 
{\it Proof.} Indeed, $F$ can be thought of  as   a map from ${\Bbb R}^n$ to ${\Bbb R}P^n$. Hence 
$Ker~F= \{\lambda_1 : \lambda_1>0\}\cong (0,\infty)$. The second part of lemma is evident.  
$\square$



\medskip
Setting $n=6g-6$ (respectively $n=2$) for $g\ge 2$ (respectively $g=1$) in 
lemmas \ref{lm5}-\ref{lm7}, one gets the
conclusion of theorem \ref{thm1}.
$\square$






%**************************************************************************
\subsection{Proof of theorem 2}
%***************************************************************************
As before, let $W$ denote  the set of toric $AF$-algebras of genus $g\ge 2$. 
Let $G$ be a finitely generated group and  $G\times W\to W$ be an action of $G$ on $W$ 
by  the stable isomorphisms of toric $AF$-algebras; in other words, 
 $\gamma ({\Bbb A}_{\theta})\otimes {\cal K}\cong
{\Bbb A}_{\theta}\otimes {\cal K}$  for all $\gamma\in G$ and all 
${\Bbb A}_{\theta}\in W$.   The following preparatory lemma will be 
important.
%**********************************************************************
\begin{lem}\label{lm2}
For each ${\Bbb A}_{\theta}\in W$,  there exists a 
representation $\rho_{{\Bbb A}_{\theta}}: G\to GL_{6g-6}({\Bbb Z})$.  
\end{lem}
%**********************************************************************
{\it Proof.}
The proof of  lemma is based on the following well known criterion
of the stable isomorphism for the $AF$-algebras: a pair of such algebras
${\Bbb A}_{\theta}, {\Bbb A}_{\theta'}$ are stably isomorphic if and only
if their Bratteli diagrams coincide, except (possibly) a finite part
of the diagram, see \cite{E}, Theorem 2.3. (Note, that the order isomorphism
between  the dimension groups mentioned in the original text, can be reformulated 
in the language of the Bratteli diagrams as stated.)  




Let $G$ be a finitely generated
group on the generators  $\{\gamma_1, \dots, \gamma_m\}$ and ${\Bbb A}_{\theta}\in W$.
Since $G$ acts on the toric $AF$-algebra ${\Bbb A}_{\theta}$ by stable isomorphisms,
the toric $AF$-algebras ${\Bbb A}_{\theta_1}:=\gamma_1({\Bbb A}_{\theta}),\dots,   
{\Bbb A}_{\theta_m}:=\gamma_m({\Bbb A}_{\theta})$ are stably isomorphic to 
${\Bbb A}_{\theta}$; moreover, by transitivity, they are also pairwise stably isomorphic.
Therefore, the Bratteli diagrams of ${\Bbb A}_{\theta_1},\dots, {\Bbb A}_{\theta_m}$ 
 coincide everywhere except, possibly, some finite parts.  
We shall denote by ${\Bbb A}_{\theta_{\max}}\in W$
a toric $AF$-algebra, whose Bratteli diagram is the maximal common part
of the Bratteli diagrams of ${\Bbb A}_{\theta_i}$ for $1\le i\le m$;
such a choice is unique and defined correctly because the set $\{{\Bbb A}_{\theta_i}\}$
is a finite set.  By the definition of a toric $AF$-algebra, the vectors 
$\theta_i=(1,\theta_1^{(i)},\dots,\theta_{6g-7}^{(i)})$
are related to the vector $\theta_{\max}=(1, \theta_1^{(\max)},\dots,\theta_{6g-7}^{(\max)})$ 
by the formula:
%*******************************************************************
$$
\left(\matrix{1\cr \theta_1^{(i)}\cr\vdots\cr\theta_{6g-7}^{(i)}} \right)
=\underbrace{
\left(\matrix{0 &  0 & \dots & 0 & 1\cr
              1 &  0 & \dots & 0 & b_1^{(1)(i)}\cr
              \vdots &\vdots & &\vdots &\vdots\cr
              0 &  0 & \dots & 1 & b_{6g-7}^{(1)(i)}}\right)
\dots 
\left(\matrix{0 &  0 & \dots & 0 & 1\cr
              1 &  0 & \dots & 0 & b_1^{(k)(i)}\cr
              \vdots &\vdots & &\vdots &\vdots\cr
              0 &  0 & \dots & 1 & b_{6g-7}^{(k)(i)}}\right)
}_{A_i}
\left(\matrix{1\cr \theta^{(\max)}_1\cr\vdots\cr\theta^{(\max)}_{6g-7}} \right)
$$
%***************************************************************** 
The above expression can be written in the matrix form $\theta_i=A_i\theta_{\max}$, where 
$A_i\in GL_{6g-6}({\Bbb Z})$. Thus, one gets a matrix representation of the
generator $\gamma_i$,  given by the formula $\rho_{{\Bbb A}_{\theta}}(\gamma_i):=A_i$.
The map $\rho_{{\Bbb A}_{\theta}}: G\to GL_{6g-6}({\Bbb Z})$ extends to the rest of the group $G$
 via its values on the generators;  namely,  for every $g\in G$ one sets $\rho_{{\Bbb A}_{\theta}}(g)= A_1^{k_1}\dots A_m^{k_m}$,
whenever $g=\gamma_1^{k_1}\dots \gamma_m^{k_m}$. It is verified (by induction), that  the map
$\rho_{{\Bbb A}_{\theta}}: G\to GL_{6g-6}({\Bbb Z})$ is a homomorphism, since 
$\rho_{{\Bbb A}_{\theta}}(g_1g_2)=\rho_{{\Bbb A}_{\theta}}(g_1)\rho_{{\Bbb A}_{\theta}}(g_2)$
for $\forall g_1,g_2\in G$. 
  Lemma \ref{lm2} follows.
$\square$ 








\medskip
Let  $W_{aper}\subset W$ be a set consisting  of the toric $AF$-algebras, whose  Bratteli diagrams
do not contain periodic (infinitely repeated) blocks;  these are known as  non-stationary toric  $AF$-algebras
and they are generic in the set $W$ endowed with the natural topology. 
We call  the action of $G$  on the toric $AF$-algebra  ${\Bbb A}_{\theta}\in W$
{\it free}, if   $\gamma ({\Bbb A}_{\theta})={\Bbb A}_{\theta}$ implies $\gamma=Id$.
%**********************************************************************
\begin{lem}\label{lm3}
Let ${\Bbb A}_{\theta}\in W_{aper}$  and  $G$ be  
free on the ${\Bbb A}_{\theta}$.  Then  $\rho_{{\Bbb A}_{\theta}}$ is a 
faithful representation.
\end{lem}
%**********************************************************************
{\it Proof.}
Since the action of $G$ is free, to prove that  $\rho_{{\Bbb A}_{\theta}}$  is faithful, 
it remains  to show, that in the formula  $\theta_i=A_i\theta_{\max}$, it holds
$A_i=I$, if and only if,  $\theta_i=\theta_{\max}$, where $I$
is the unit matrix.   Indeed,  it is immediate that $A_i=I$ implies $\theta_i=\theta_{\max}$.
Suppose now that  $\theta_i=\theta_{\max}$ and, let to the contrary, $A_i\ne I$. 
One gets $\theta_i=A_i \theta_{\max}=\theta_{\max}$.  Such an equation has  a non-trivial solution, 
if and only if,  the vector $\theta_{\max}$ has a periodic 
Jacobi-Perron fraction;  the period of  such a fraction  is given by the matrix $A_i$. This 
is impossible, since it has been assumed, that ${\Bbb A}_{\theta_{\max}}\in W_{aper}$.
The contradiction finishes the proof of lemma \ref{lm3}.
$\square$


 
\medskip
Let $G=Mod~(X)$, where $X$ is a surface of genus $g\ge 2$. The group $G$
is finitely generated \cite{Deh1}; it  acts on the Teichm\"uller space $T(g)$
by isomorphisms of the Riemann surfaces. Moreover, the action of $G$ is free on a 
generic set, $U\subset T(g)$,  consisting of the Riemann surfaces with the trivial group 
of the automorphisms. Recall,  that there exists a functor  $F: V\to W$  between the Riemann surfaces and 
toric $AF$-algebras, see theorem \ref{thm1}. The following lemma is true.
%**********************************************************************
\begin{lem}\label{lm4}
The pre-image  $F^{-1}(W_{aper})$ is a generic set in the space  $T(g)$.
\end{lem}
%**********************************************************************
{\it Proof.}
Note,  that the set of stationary toric $AF$-algebras is a countable set.
The functor $F$ is a surjective map,  which is continuous with respect to the natural topology 
on the sets $V$ and $W$.  Therefore,  the pre-image of the complement of a countable set is a generic set. 
$\square$



\medskip
To finish the proof,  consider the set $U\cap F^{-1}(W_{aper})$;
this set is a non-empty set, since it is the intersection of  two generic subsets of $T(g)$.
Let $R$ be a point (a Riemann surface) in the above set.  In  view of theorem \ref{thm1},
group $G$ acts on the toric $AF$-algebra ${\Bbb A}_{\theta}=F(R)$ by the stable
isomorphisms.  By the construction, the action is free and ${\Bbb A}_{\theta}\in W_{aper}$.
In view of lemma \ref{lm3},  one gets a faithful representation $\rho=\rho_{{\Bbb A}_{\theta}}$
of the group $G=Mod~(X)$ into the matrix group $GL_{6g-6}({\Bbb Z})$.
Theorem \ref{thm2} follows.
$\square$





%**************************************************************************
\section{Appendix}
%***************************************************************************
This section is a brief review of the results used to prove
theorems \ref{thm1} and \ref{thm2}.  For a systematic account,  we refer the reader
to \cite{B}, \cite{E}, \cite{HuMa1} and \cite{Thu2}. 



%*********************************************************************
\subsection{Measured foliations and $T(g)$}
%*********************************************************************
{\bf A. The measured foliations.}  A measured foliation, ${\cal F}$, on a surface $X$
is a  partition of $X$ into the singular points $x_1,\dots,x_n$ of
order $k_1,\dots, k_n$ and regular leaves (1-dimensional submanifolds). 
On each  open cover $U_i$ of $X-\{x_1,\dots,x_n\}$ there exists a non-vanishing
real-valued closed 1-form $\phi_i$  such that: 
(i)  $\phi_i=\pm \phi_j$ on $U_i\cap U_j$;
(ii) at each $x_i$ there exists a local chart $(u,v):V\to {\Bbb R}^2$
such that for $z=u+iv$, we have $\phi_i=Im~(z^{k_i\over 2}dz)$ on
$V\cap U_i$ for some branch of $z^{k_i\over 2}$. 
The pair $(U_i,\phi_i)$ is called an atlas for the measured foliation ${\cal F}$.
Finally, a measure $\mu$ is assigned to each segment $(t_0,t)\in U_i$, which is  transverse to
the leaves of ${\cal F}$, via the integral $\mu(t_0,t)=\int_{t_0}^t\phi_i$. The 
measure is invariant along the leaves of ${\cal F}$, hence the name. 



\smallskip\noindent
{\bf B. The Hubbard-Masur embedding.}
 Let $S$ be a Riemann surface, and $q\in H^0(S,\Omega^{\otimes 2})$ a holomorphic
quadratic differential on $S$. The lines $Re~q=0$ and $Im~q=0$ define a pair
of measured foliations on $S$, which are transversal to each other outside the set of 
singular points. The set of singular points is common to both foliations and coincides
with the zeroes of $q$. The above measured foliations are said to represent the  vertical and horizontal 
 trajectory structure  of $q$, respectively. 
 Let $T(g)$ be the Teichm\"uller space of the topological surface $X$ of genus $g\ge 1$,
i.e. the space of the complex structures on $X$. 
Consider the vector bundle $p: Q\to T(g)$ over $T(g)$ whose fiber above a point 
$S\in T_g$ is the vector space $H^0(S,\Omega^{\otimes 2})$.   
Given non-zero $q\in Q$ above $S$, we can consider horizontal measured foliation
${\cal F}_q\in \Phi_X$ of $q$, where $\Phi_X$ denotes the space of equivalence
classes of measured foliations on $X$. If $\{0\}$ is the zero section of $Q$,
the above construction defines a map $Q-\{0\}\longrightarrow \Phi_X$. 
For any ${\cal F}\in\Phi_X$, let $E_{\cal F}\subset Q-\{0\}$ be the fiber
above ${\cal F}$. In other words, $E_{\cal F}$ is a subspace of the holomorphic 
quadratic forms whose horizontal trajectory structure coincides with the 
measured foliation ${\cal F}$. 
Note that, if ${\cal F}$ is a measured foliation with the simple zeroes (a generic case),  
then $E_{\cal F}\cong {\Bbb R}^n - 0$, while $T(g)\cong {\Bbb R}^n$, where $n=6g-6$ if
$g\ge 2$ and $n=2$ if $g=1$.
%************************************************
\begin{thm}\label{thm3}
{\bf (\cite{HuMa1})}
The restriction of $p$ to $E_{\cal F}$ defines a homeomorphism (an embedding)
$h_{\cal F}: E_{\cal F}\to T(g)$
%**********************************************************************************
\footnote{To be precise, the theorem mentions a {\it local} homeomorphism
$h_{\cal F}: E_{\cal F}\to T(g)$,
ibid, p.222. However, since $T(g)$ is simply connected, $h_{\cal F}$ extends to
a {\it global} homeomorphism between the two spaces. Indeed, let
$\lambda\in  E_{\cal F}$. 
It is easy to see, that $\lambda=0$ corresponds to the distinguished
point $S\in T(g)$, while $\lambda=\infty$ represent the boundary
of the space $T(g)$; thus, every local chart $|\lambda|<C$ extends 
to the global chart $|\lambda|<\infty$.}
%**********************************************************************************
.
\end{thm}
%*************************************************
The Hubbard-Masur result implies that the measured foliations  parametrize  
the space $T(g)-\{pt\}$, where $pt= h_{\cal F}(0)$.
Indeed, denote by  ${\cal F}'$ a vertical trajectory structure of  $q$. Since ${\cal F}$
and ${\cal F}'$ define $q$, and ${\cal F}=Const$ for all $q\in E_{\cal F}$, one gets a homeomorphism 
between $T(g)-\{pt\}$ and  $\Phi_X$, where $\Phi_X\cong {\Bbb R}^n-0$ is the space of 
equivalence classes of the measured foliations ${\cal F}'$ on $X$. 
Note that the above parametrization depends on a foliation ${\cal F}$.
However, there exists a unique canonical homeomorphism $h=h_{\cal F}$
as follows. Let $Sp ~(S)$ be the length spectrum of the Riemann surface
$S$ and $Sp ~({\cal F}')$  the set of positive reals $\inf \mu(\gamma_i)$,
where $\gamma_i$ runs over all simple closed curves, which are transverse 
to the foliation ${\cal F}'$. A canonical homeomorphism 
$h=h_{\cal F}: \Phi_X\to T(g)-\{pt\}$ is defined by the formula
$Sp ~({\cal F}')= Sp ~(h_{\cal F}({\cal F}'))$ for $\forall {\cal F}'\in\Phi_X$. 
Thus, the following corollary is true.
%********************************************************
\begin{cor}\label{cor1}
There exists a unique homeomorphism $h:\Phi_X\to T(g)-\{pt\}$.
\end{cor}
%****************************************************



\smallskip\noindent
{\bf C. The Douady-Hubbard coordinates.}
Recall that $\Phi_X$ is the space of equivalence classes of measured
foliations on the topological surface $X$. Following Douady and Hubbard
\cite{DoHu1}, we consider a coordinate system on $\Phi_X$,
suitable for the proof of theorem \ref{thm1}. 
For clarity, let us make a generic assumption that $q\in H^0(S,\Omega^{\otimes 2})$
is a non-trivial holomorphic quadratic differential with only simple zeroes. 
We wish to construct a Riemann surface of $\sqrt{q}$, which is a double cover
of $S$ with ramification over the zeroes of $q$. Such a surface, denoted by
$\widetilde S$, is unique and has advantage of carrying a holomorphic
differential $\omega$, such that $\omega^2=q$. We further denote by 
$\pi:\widetilde S\to S$ the covering projection. The vector space
$H^0(\widetilde S,\Omega)$ splits into the direct sum
$H^0_{even}(\widetilde S,\Omega)\oplus H^0_{odd}(\widetilde S,\Omega)$
in view of  the involution $\pi^{-1}$ of $\widetilde S$, and
the vector space $H^0(S,\Omega^{\otimes 2})\cong H^0_{odd}(\widetilde S,\Omega)$.
Let $H_1^{odd}(\widetilde S)$ be the odd part of the homology of $\widetilde S$
relative to  the zeroes of $q$.   Consider the pairing
$H_1^{odd}(\widetilde S)\times H^0(S, \Omega^{\otimes 2})\to {\Bbb C}$,
defined by the integration  $(\gamma, q)\mapsto \int_{\gamma}\omega$. 
We shall take the associated map
$\psi_q: H^0(S,\Omega^{\otimes 2})\to Hom~(H_1^{odd}(\widetilde S); {\Bbb C})$
and let $h_q= Re~\psi_q$. 
%************************************************************
\begin{thm}\label{thm4}
{\bf (\cite{DoHu1}, p.176)}
The map $h_q: H^0(S, \Omega^{\otimes 2})\longrightarrow Hom~(H_1^{odd}(\widetilde S); {\Bbb R})$
is an ${\Bbb R}$-isomorphism. 
\end{thm}
%******************************************************* 
Since  each  ${\cal F}\in \Phi_X$ is the  vertical foliation 
$Re~q=0$ for a $q\in H^0(S, \Omega^{\otimes 2})$, the Douady-Hubbard lemma
implies that $\Phi_X\cong Hom~(H_1^{odd}(\widetilde S); {\Bbb R})$.
By  formulas for the relative homology, 
one finds that $H_1^{odd}(\widetilde S)\cong {\Bbb Z}^{n}$,
where $n=6g-6$ if $g\ge 2$ and $n=2$ if $g=1$.     
Finally, each $h\in Hom~({\Bbb Z}^{n}; {\Bbb R})$ is given
by the reals  $\lambda_1=h(e_1),\dots, \lambda_{n}=h(e_{n})$,
where $(e_1,\dots, e_{n})$ is a basis in ${\Bbb Z}^{n}$.
The numbers   $(\lambda_1,\dots,\lambda_{n})$ are the coordinates in the space $\Phi_X$
and, in view of the corollary \ref{cor1}, in  the Teichm\"uller space $T(g)$. 










%**********************************************************************
\subsection{The Jacobi-Perron continued fraction}
%*********************************************************************
{\bf A. The regular continued fractions.}
Let $a_1,a_2\in {\Bbb N}$ such that $a_2\le a_1$. Recall that the greatest common
divisor of $a_1$ and $a_2$, $GCD(a_1,a_2)$, can be determined from the Euclidean algorithm:
%*******************************************************************
$$
\left\{
\begin{array}{cc}
a_1 &= a_2b_1 +r_3\nonumber\\
a_2 &= r_3b_2 +r_4\nonumber\\
r_3 &= r_4b_3 +r_5\nonumber\\
\vdots & \nonumber\\
r_{k-3} &= r_{k-2}b_{k-1}+r_{k-1}\nonumber\\
r_{k-2} &= r_{k-1}b_k,
\end{array}
\right.
$$
%*****************************************************************
where $b_i\in {\Bbb N}$ and $GCD(a_1,a_2)=r_{k-1}$. 
The Euclidean algorithm can be written as the regular continued 
fraction
%*************************************************************
$$
\theta={a_1\over a_2}=b_1+{1\over\displaystyle b_2+
{\strut 1\over\displaystyle +\dots+ {1\over b_k}}}
=[b_1,\dots b_k].
$$
%************************************************************
If $a_1$ and $a_2$ are non-commensurable, in the sense that $\theta\in {\Bbb R}-{\Bbb Q}$,
then the Euclidean algorithm never stops and $\theta=[b_1, b_2, \dots]$. Note that the regular  
continued fraction can be written in the matrix form:
%*********************************************************************
$$
\left(\matrix{1\cr \theta}\right)=
\lim_{k\to\infty} \left(\matrix{0 & 1\cr 1 & b_1}\right)\dots
\left(\matrix{0 & 1\cr 1 & b_k}\right)
\left(\matrix{0\cr 1}\right). 
$$
%********************************************************************



\medskip\noindent
{\bf B. The Jacobi-Perron continued fractions.}
The Jacobi-Perron algorithm and connected (multidimensional) continued 
fraction generalizes the Euclidean algorithm to the case $GCD(a_1,\dots,a_n)$
when $n\ge 2$.  Specifically, let $\lambda=(\lambda_1,\dots,\lambda_n)$,
$\lambda_i\in {\Bbb R}-{\Bbb Q}$ and  $\theta_{i-1}={\lambda_i\over\lambda_1}$  with
$1\le i\le n$.   The continued fraction 
%**************************************************************
$$
\left(\matrix{1\cr \theta_1\cr\vdots\cr\theta_{n-1}} \right)=
\lim_{k\to\infty} 
\left(\matrix{0 &  0 & \dots & 0 & 1\cr
              1 &  0 & \dots & 0 & b_1^{(1)}\cr
              \vdots &\vdots & &\vdots &\vdots\cr
              0 &  0 & \dots & 1 & b_{n-1}^{(1)}}\right)
\dots 
\left(\matrix{0 &  0 & \dots & 0 & 1\cr
              1 &  0 & \dots & 0 & b_1^{(k)}\cr
              \vdots &\vdots & &\vdots &\vdots\cr
              0 &  0 & \dots & 1 & b_{n-1}^{(k)}}\right)
\left(\matrix{0\cr 0\cr\vdots\cr 1} \right),
$$
%**************************************************************
where $b_i^{(j)}\in {\Bbb N}\cup\{0\}$, is called the {\it Jacobi-Perron
algorithm (JPA)}. Unlike the regular continued fraction algorithm,
the JPA may diverge for certain vectors $\lambda\in {\Bbb R}^n$. However, 
for points of a generic subset of ${\Bbb R}^n$, the JPA converges. 
The convergence of the JPA algorithm can be characterized in terms of
the measured foliations. Let ${\cal F}\in\Phi_X$ be a measured foliation
on the surface $X$ of genus $g\ge 1$. Recall that ${\cal F}$ is called uniquely
ergodic if every invariant measure of ${\cal F}$ is a multiple
of the Lebesgue measure. By the Masur-Veech theorem, there exists
a generic subset $V\subset \Phi_X$ such that each ${\cal F}\in V$
is uniquely ergodic \cite{Mas1}, \cite{Vee1}.
We let $\lambda=(\lambda_1,\dots,\lambda_{n})$ be the vector with
coordinates $\lambda_i=\mu ({\gamma_i})$, where $\gamma_i\in H_1^{odd}(\widetilde S)$. 
By an abuse of notation, we shall say that $\lambda\in V$. 
By a duality  between the measured foliations and the interval exchange transformations \cite{Mas1}, 
there is the following characterization of convergence of the JPA.
%********************************************************
\begin{thm}\label{thm5}
{\bf (\cite{Bau1})}
The JPA converges if and only if $\lambda\in V\subset {\Bbb R}^{n}$.  
\end{thm}
%******************************************************
  





%**************************************************************************
\subsection{$AF$-algebras}
%***************************************************************************
{\bf A. The $C^*$-algebras.}
By a {\it $C^*$-algebra} one understands the Banach
algebra with an involution. Namely, a $C^*$-algebra
$A$ is an algebra over the complex numbers $\Bbb C$ with a norm $a\mapsto ||a||$
and an involution $a\mapsto a^*, a\in A$, such that $A$ is
complete with the respect to the norm, and such that
$||ab||\le ||a||~||b||$ and $||a^*a||=||a||^2$ for every
$a,b\in A$. If  $A$ is commutative, then the Gelfand
theorem says that $A$ is isometrically $*$-isomorphic
to the $C^*$-algebra $C_0(X)$ of the continuous complex-valued
functions on a locally compact Hausdorff space $X$.
For otherwise, the algebra $A$ represents a noncommutative topological
space.



\medskip\noindent
{\bf B.  The stable isomorphisms of $C^*$-algebras.} 
Let $A$ be a $C^*$-algebra deemed as a noncommutative
topological space. One can ask, when two such topological spaces
$A,A'$ are homeomorphic? To answer the question, let us recall
the topological $K$-theory.  If $X$ is a (commutative) topological
space, denote by $V_{\Bbb C}(X)$ an abelian monoid consisting
of the isomorphism classes of the complex vector bundles over $X$
endowed with the Whitney sum. The abelian monoid $V_{\Bbb C}(X)$
can be made to an abelian group, $K(X)$, using the Grothendieck
completion. The covariant functor $F: X\to K(X)$ is known to
map the homeomorphic topological spaces $X,X'$ to the isomorphic
abelian groups $K(X), K(X')$. Let  $A,A'$ be the $C^*$-algebras. If one wishes to
define a homeomorphism between the noncommutative topological spaces $A$ and $A'$, 
it will suffice to define an isomorphism between the abelian monoids $V_{\Bbb C}(A)$
and $V_{\Bbb C}(A')$ as suggested by the topological $K$-theory. 
The role of the complex vector bundle of the degree $n$ over the 
$C^*$-algebra $A$ is played by a $C^*$-algebra $M_n(A)=A\otimes M_n$,
i.e. the matrix algebra with the entries in $A$.  The abelian monoid
$V_{\Bbb C}(A)=\cup_{n=1}^{\infty} M_n(A)$ replaces the monoid 
$V_{\Bbb C}(X)$ of the topological $K$-theory. Therefore, 
the noncommutative topological spaces $A,A'$ are homeomorphic,
if the abelian monoids $V_{\Bbb C}(A)\cong V_{\Bbb C}(A')$ are isomorphic. 
The latter equivalence is called a {\it stable isomorphism}
of the $C^*$-algebras $A$ and $A'$ and is formally written as 
$A\otimes {\cal K}\cong A'\otimes {\cal K}$, where 
${\cal K}=\cup_{n=1}^{\infty}M_n$ is the $C^*$-algebra of compact
operators.  Roughly speaking, the stable isomorphism between
the $C^*$-algebras  means that they  are
homeomorphic as the noncommutative topological spaces.    



\medskip\noindent
{\bf C. The $AF$-algebras.}
An {\it $AF$-algebra}  (approximately finite $C^*$-algebra) is defined to
be the  norm closure of an ascending sequence of the finite dimensional
$C^*$-algebras $M_n$'s, where  $M_n$ is the $C^*$-algebra of the $n\times n$ matrices
with the entries in ${\Bbb C}$. Here the index $n=(n_1,\dots,n_k)$ represents
a semi-simple matrix algebra $M_n=M_{n_1}\oplus\dots\oplus M_{n_k}$.
The ascending sequence mentioned above  can be written as 
%***********************************************************
$M_1\buildrel\rm\varphi_1\over\longrightarrow M_2
   \buildrel\rm\varphi_2\over\longrightarrow\dots,
$
%****************************************************
where $M_i$ are the finite dimensional $C^*$-algebras and
$\varphi_i$ the homomorphisms between such algebras.  The set-theoretic limit
$A=\lim M_n$ has a natural algebraic structure given by the formula
$a_m+b_k\to a+b$; here $a_m\to a,b_k\to b$ for the
sequences $a_m\in M_m,b_k\in M_k$.  
The homomorphisms $\varphi_i$ can be arranged into  a graph as follows. 
Let  $M_i=M_{i_1}\oplus\dots\oplus M_{i_k}$ and 
$M_{i'}=M_{i_1'}\oplus\dots\oplus M_{i_k'}$ be 
the semi-simple $C^*$-algebras and $\varphi_i: M_i\to M_{i'}$ the  homomorphism. 
One has the two sets of vertices $V_{i_1},\dots, V_{i_k}$ and $V_{i_1'},\dots, V_{i_k'}$
joined by the $a_{rs}$ edges, whenever the summand $M_{i_r}$ contains $a_{rs}$
copies of the summand $M_{i_s'}$ under the embedding $\varphi_i$. 
As $i$ varies, one obtains an infinite graph called a {\it Bratteli diagram} of the
$AF$-algebra. The Bratteli diagram defines a unique  $AF$-algebra.




\medskip\noindent
{\bf D. The stationary $AF$-algebras.}
If the homomorphisms $\varphi_1 =\varphi_2=\dots=Const$ in the definition of 
the $AF$-algebra $A$,  the $AF$-algebra $A$ is called {\it stationary}. 
The Bratteli diagram of a stationary $AF$-algebra looks like a periodic 
graph  with the incidence matrix $A=(a_{rs})$ repeated over and over again. 
Since  matrix $A$ is a non-negative integer matrix, one can take a power of
$A$ to obtain a strictly positive integer matrix -- which we always assume 
to be the case.  The stationary $AF$-algebra has a non-trivial group
of the automorphisms \cite{E}, Ch.6. 














  

%**************************************************************************
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\end{thebibliography}
%**********************************************************

\vskip1cm


\textsc{The Fields Institute for Mathematical Sciences, Toronto, ON, Canada,  
E-mail:} {\sf igor.v.nikolaev@gmail.com} 


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{\it Current address: 101-315 Holmwood Ave., Ottawa, ON, Canada, K1S 2R2}  

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math0111173.extracted_bib
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