Hyperbolic geometry, continued fractions and classification of the finitely generated totally ordered dimension groups
Igor ~Nikolaev \footnote{Partially supported by NSERC.}
We classify polycyclic totally ordered dimension groups, i.e. dimension groups generated by dense embeddings of the lattice ${\Bbb Z}^{n}$ in the real line ${\Bbb R}$ . Our method is based on geometry of simple geodesics on the modular surface of genus $g\ge2$. The main theorem says that isomorphism classes of the polycyclic totally ordered dimension group are bijective with the reals $\alpha$ modulo the action of the group $GL(2,{\Bbb Z})$. The result is an extension of the Effros-Shen classification of the dicyclic dimension groups. \vspace{7mm} {Key words and phrases: dimension group, geodesic lamination, continued fraction} \vspace{5mm} {AMS (MOS) Subj. Class.: 46L85}
math0310400.tex
%******************************************************************    
%%%%\documentclass[12pt]{article}    
\documentclass{article}



\input{amssym.def} 
\input{amssym.tex}



\begin{document}      
%**********************************************    
\title{Hyperbolic geometry, continued fractions
      and classification of the finitely generated totally ordered
      dimension groups}    
%***********************************************    


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



\date{}     
%\date{March 20, 2003}    
 \maketitle    
    



\newtheorem{thm}{Theorem}    
\newtheorem{lem}{Lemma}    
\newtheorem{dfn}{Definition}    
\newtheorem{rmk}{Remark}    
\newtheorem{cor}{Corollary}    
\newtheorem{prp}{Proposition}    
\newtheorem{exm}{Example}    
    
%******************************************************************    
 \begin{abstract}    
We  classify polycyclic totally ordered dimension groups, i.e. dimension groups 
generated by dense embeddings of the lattice  ${\Bbb Z}^{n}$ in the real line ${\Bbb R}$ . Our method is based on
geometry of simple geodesics on the modular surface of genus $g\ge2$.
The main theorem says that isomorphism classes of the  polycyclic totally ordered dimension group 
are bijective with the reals $\alpha$  modulo the action of the group  $GL(2,{\Bbb Z})$. 
The result is an extension of the Effros-Shen classification of the dicyclic dimension
groups.  





   
\vspace{7mm}    
    
{\it Key words and phrases:  dimension group, geodesic lamination,  continued fraction} 



\vspace{5mm}
{\it AMS (MOS) Subj. Class.:  46L85}
\end{abstract}







%**************************************************************************
\section{Introduction}
%***************************************************************************
The dimension of a subspace of the euclidean space ${\Bbb R}^n$  can be 1, 2, 3 or more.
It is no longer true that dimension of any non-trivial subspace of the euclidean
space ${\Bbb R}^{\infty}$ is a positive integer.  It was discovered by von Neumann 
that there exist linear subspaces in ${\Bbb R}^{\infty}$ of a non-integer 
(continuous) dimension. In fact, there exists a dimension function on projections
in a von Neumann algebra, which takes on any real value in the interval 
$[0,1]$. 




Unlike von Neumann algebras, the  dimension function on the  projections of a  
$C^*$-algebra, $A$,   takes value in an abelian group, $K_0(A)$,  rather than in ${\Bbb R}$
\cite{E}, Ch. 1.  The range of dimension function in $K_0(A)$ is known as a {\it dimension
group} of $A$. The classification of such groups is a difficult open problem. 



In \cite{EfS} Effros and Shen classified the dicyclic dimension groups,
i.e. dimension groups inside the  abelian group  ${\Bbb Z}^2$.
The  result says that each dicyclic group can be assigned a
positive irrational number $\alpha$, defined up to the action
of matrix group $GL(2,{\Bbb Z})$, and such that if



\bigskip\noindent
%***************************************************************************
\begin{equation}\label{eq1}
\alpha=a_0+{1\over\displaystyle a_1+
{\strut 1\over\displaystyle a_2\displaystyle +\dots}}=[a_0,a_1,a_2,\dots],   
\end{equation}
%*****************************************************************

\medskip\noindent
is a regular continued fraction of $\alpha$,  then one gets a representation   
of the dicyclic group via simplicial dimension groups:
%**************************************************************
\begin{equation}\label{eq2}
{\Bbb Z}^2\buildrel\rm 
\left(\small\matrix{0 & 1\cr 1 & a_0}\right)
\over\longrightarrow {\Bbb Z}^2
\buildrel\rm 
\left(\small\matrix{0 & 1\cr 1 & a_1}\right)
\over\longrightarrow
{\Bbb Z}^2\buildrel\rm 
\left(\small\matrix{0 & 1\cr 1 & a_2}\right)
\over\longrightarrow \dots
\end{equation}
%*************************************************************** 
see Theorem 3.2 of \cite{EfS}.
The  irrational $\alpha$ is a {\it slope}  of the  straight line in the plane, which is
the universal cover of  a two-dimensional torus, see \cite{EfS} Theorem 2.1. 
The range of dimension function (an order) in ${\Bbb Z}^2$ is described by the cone 
$P_{\alpha}=\{(k,l)\in {\Bbb Z}^2~|~\alpha k+l\ge 0\}$,  and it is shown that every 
dicyclic dimension group  arises in this way. 





An objective of present note is  similar classification of the polycyclic
dimension groups, i.e. the dimension groups inside the abelian group ${\Bbb Z}^n$.   
Recall that a universal cover for the surfaces of genus $g\ge 2$ is no longer the euclidean plane
but a hyperbolic plane, ${\Bbb H}$. It is well known that a substitution  for
the straight lines in ${\Bbb H}$ is given by the simple geodesics, i.e.
geodesics with no self-crossing or self-tangent points. Thus, 
to classify the polycyclic dimension groups, one needs: 

\medskip
(i) to define a dimension  group, $G$,  coming from the simple
geodesic, $\gamma$,  on a Riemann surface of genus $g\ge2$;   

\smallskip
(ii) to introduce a slope (a real number $\alpha$) of $\gamma$ on the surface; 
 
 
\smallskip 
 (iii) to construct  a simplicial   approximation   of $G$ 
in terms of the given slope $\alpha$.  



\smallskip
The realization of (i) -- (iii) is as follows. The closure, $\bar\gamma$,
of a simple non-periodic geodesic $\gamma$ consists of a continuum 
of disjoint non-periodic geodesics known as a {\it geodesic lamination}
$\lambda$ \cite{CaB}. Let $|\lambda|$ be the total number of the principal
regions of $\lambda$ ({\it ibid.}, p.60) and $n=2g+|\lambda|-1$, where
$g$ is the genus of Riemann surface carrying the lamination $\lambda$. 
It is well known that the set of invariant transversal measures of $\lambda$
is a convex compact set, $\Delta_{k-1}$,  of dimension $1\le k\le\left[{n\over 2}\right]$,
where $[\bullet]$ is the integer part of a number \cite{Sat}.  By $G_{\lambda}$
we understand a dimension group inside the abelian group ${\Bbb Z}^n$,
whose state space $S(G_{\lambda})$ is isomorphic to $\Delta_{k-1}$. 
The assignment of $G_{\lambda}$ to $\lambda$ is unique \cite{G}. 
To implement (ii), one needs to restrict to the Legendre laminations 
on a modular surface $X(N)={\Bbb H}/\Gamma(N)$, where $\Gamma(N)$
is the principal congruence group. The slope $\alpha$ of $\lambda$
is given by a regular continued fraction generated by the periodic      
geodesics, which approximate $\lambda$ in the space of geodesic 
laminations on $X(N)$. Finally, to realize item (iii), one needs
to restrict to a generic case $k=1$ (i.e. totally ordered dimension
groups). The condition secures the convergence of a Jacobi-Perron
continued fraction attached to lamination $\lambda_{\alpha}$ 
and one can apply the known theorem of Effros and Shen \cite{EfSh2}
on the unimodular dimension groups.  



To formulate our main results, let ${\cal F}_{\alpha}$ be a foliation
on the surface $X=X(N)$ obtained by a blow-down of the lamination
$\lambda_{\alpha}$ (see appendix). For simplicity, we let ${\cal F}_{\alpha}$
be given by trajectories of a closed one-form, $\omega_{\alpha}$, on $X$. 
The periods of $\omega_{\alpha}$ in a basis $\{\gamma_1,\dots,\gamma_n\}$
in the group  $H_1(X, Sing~\omega_{\alpha}; {\Bbb Z})$ will be denoted by
$\lambda_i=\int_{\gamma_i}\omega_{\alpha}$.  Let $\theta_i=\lambda_{i-1}/\lambda_1$ 
and $\theta=(\theta_1,\dots,\theta_{n-1})$. Finally, let  the Jacobi-Perron continued 
fraction of $\theta$ be:
%**********************************************************************************
\begin{equation}\label{eq3}
\left(\matrix{1\cr \theta}\right)=
\lim_{k\to\infty} \left(\matrix{0 & 1\cr I & b_0}\right)\dots
\left(\matrix{0 & 1\cr I & b_k}\right)
\left(\matrix{0\cr {\Bbb I}}\right),
\end{equation}
%************************************************************ 
where $I$ is the unit matrix, $b_i=(b^{(i)}_1,\dots, b^{(i)}_{n})^T$ a 
vector of  the non-negative integers and ${\Bbb I}=(0,\dots, 0, 1)^T$.
A summary of our results can be formulated as follows.
%*****************************************************************
\begin{thm}
Each finitely generated totally ordered dimension group $G$
can be indexed by a positive irrational number $\alpha\in U$,
where $U$ is a generic subset of ${\Bbb R}$. The parametrization, $G_{\alpha}$,
has the following properties:


\medskip
(i) $G_{\alpha}$ and $G_{\alpha'}$ are order-isomorphic if and only 
if $\alpha'=\alpha ~mod ~GL(2, {\Bbb Z})$;


\smallskip
(ii) $G_{\alpha}$  is the limit of the following  simplicial dimension groups: 
%*************************************************************
\begin{equation}\label{eq4}
{\Bbb Z}^n
\buildrel\rm
\left(\matrix{0 & 1\cr I & b_0}\right)
\over\longrightarrow 
{\Bbb Z}^n
\buildrel\rm
\left(\matrix{0 & 1\cr I & b_1}\right)
\over\longrightarrow
{\Bbb Z}^n
\dots
\end{equation}
%**************************************************************
\end{thm}
%***********************************************************
The article is organized as follows. In section 2 we recall some useful
definitions and facts on the dimension groups. The dimension group $G_{\lambda}$ 
is introduced in section 3. The Legendre laminations are revised in section 4.
Theorem 1 is proved in section 5. Finally, in section 6 a brief account 
of geodesic laminations,  modular surfaces and the Jacobi-Perron fractions is 
given. 








\bigskip\noindent
{\sf Acknowledgments.} I wish to thank G.~Elliott for
helpful discussions.   



%**************************************************************************
\section{Dimension groups}
%***************************************************************************
We use ${\Bbb Z},{\Bbb Z}^+,{\Bbb Q}$ and ${\Bbb R}$ for the integers,
positive integers, rationals and reals, respectively and $GL(n,{\Bbb Z})$
for  the group of $n\times n$ matrices with entries in ${\Bbb Z}$
and determinant $\pm 1$. 


By an {\it ordered group} we shall mean an abelian group $G$ together
with a subset $P=G^+$ such that $P+P\subseteq P, P\cap (-P)=\{0\}$,
and $P-P=G$. We call $P$ the {\it positive cone} on $G$. We write
$a\le b$ (or $a<b$) if $b-a\in P$ (or $b-a\in P\backslash\{0\}$).


G is said to be a {\it Riesz group} if:

\medskip
(i) $g\in G$ and $ng\ge 0, n\in {\Bbb Z}^+$ implies $g\ge 0$; 

\smallskip
(ii) $u,v\le x,y$ in $G$ implies existence of $w\in G$
such that $u,v\le w\le x,y$.


\smallskip
Given ordered groups $G$ and $H$, we say that a homomorphism
$\varphi:G\to H$ is {\it positive} if $\varphi(G^+)\subseteq H^+$,
and that $\varphi:G\to H$ is an {\it order isomorphism}
if $\varphi(G^+)=H^+$.



A positive homomorphism $f:G\to {\Bbb R}$ is called a {\it state}
if $f(u)=1$, where $u\in G^+$ is an order unit of $G$.
We let $S(G)$ be the {\it state space} of $G$, i.e. the set of states
on $G$ endowed with natural topology \cite{E}. 

$S(G)$ is a compact convex subset of the vector space $Hom~(G,{\Bbb R})$.
By the Krein-Milman theorem, $S(G)$ is the closed convex hull
of its extreme points, which are called {\it pure states}.


An ordered abelian group is a {\it dimension group}
if it is order isomorphic to $\lim_{m.n\to\infty}
({\Bbb Z}^{r(m)},\varphi_{mn})$, where the ${\Bbb Z}^{r(m)}$
are  simplicially ordered groups (i.e. $({\Bbb Z}^{r(m)})^+
\cong {\Bbb Z}^+\oplus\dots\oplus {\Bbb Z}^+$), and the $\varphi_{mn}$
are positive homomorphisms. The dimension group $G$ is said to be
{\it unimodular} if $r(m)=Const=r$ and $\varphi_{mn}$
are positive isomorphisms of ${\Bbb Z}^r$. In other words, $G$ is
the limit   
%**************************************************************
\begin{equation}\label{eq5}
{\Bbb Z}^r\buildrel\rm
\varphi_0
\over\longrightarrow {\Bbb Z}^r
\buildrel\rm
\varphi_1
\over\longrightarrow
{\Bbb Z}^r\buildrel\rm
\varphi_2
\over\longrightarrow \dots,
\end{equation}
%*********************************************************************
of matrices $\varphi_k\in GL(r,{\Bbb Z}^+)$.



The Riesz groups are dimension groups and vice versa.
The Riesz groups can be viewed as  the abstract dimension groups, while
dimension groups as  a  representation  of the Riesz groups
by the infinite sequences of positive homomorphisms.






%**********************************************************************
\section{Dimension groups generated by simple geodesics}
%***********************************************************************
Let $\gamma$ be a simple non-periodic geodesic on a Riemann surface $X$
of genus $g\ge 2$. The closure $\bar\gamma$ contains a continuum of non-periodic
pairwise disjoint simple geodeisics, which form a perfect (Cantor) set on the
surface $X$. The $\bar\gamma$ is a geodesic lamination, which we shall denote
by $\lambda$. The number of principal regions of $\lambda$ will be denoted 
by $|\lambda|$. 


It is well known that $\lambda$ corresponds to a foliation, ${\cal F}_{\lambda}$,
on $X$. The ${\cal F}_{\lambda}$ is a foliation with $|\lambda|$ singular
points of the saddle type. The ${\cal F}_{\lambda}$ is obtained from $\lambda$
by a blow-down homotopy \cite{T}. 
Denote by $\Phi_X$ the space of foliations on $X$, whose singularity set coincides
with such of ${\cal F}_{\lambda}$. The coordinates of ${\cal F}_{\lambda}$
in $\Phi_X$ are given by the formula: 
%**************************************************************
\begin{equation}\label{eq6}
\lambda_i=\int_{\gamma_i}{\cal F}_{\lambda}d\mu,
\end{equation}
%*************************************************************
where $\{\gamma_1,\dots,\gamma_n\}$ is a basis in the group
$H_1(X, Sing~{\cal F}_{\lambda}; {\Bbb Z})$ and $\mu$ 
an invariant transversal measure on the leaves of foliation
${\cal F}_{\lambda}$. It follows from the formulas
for the relative homology that:  
%**********************************************************
\begin{equation}\label{eq7}
n=2g+|\lambda|-1.
\end{equation}
%************************************************************
On the other hand, it is known that the number, $k$, of the linearly
independent invariant measures of the foliation ${\cal F}_{\lambda}$
cannot exceed $\left[{n\over 2}\right]$ \cite{Sat}. In view of formula
(\ref{eq6}), each independent measure $\mu_i, 1\le i\le k$, defines 
a homomorphism ${\Bbb Z}^n\to {\Bbb R}$, whose kernel is a hyperplane
in ${\Bbb R}^n$. Thus, one obtains a dimension group, $G_{\lambda}$,
inside the abelian group ${\Bbb Z}^n$, which is bounded by the
$k$ hyperplanes corresponding to the measures $\mu_1,\dots,\mu_k$.    
%*************************************************************************
\begin{dfn}\label{dfn1}
The dimension group $G_{\lambda}$ is called a group associated to the
simple geodesic $\gamma$. 
\end{dfn}
%***********************************************************************
In the sequel, our main case will be $k=1$ (i.e. the totally ordered dimension group
$G_{\lambda}$). Note that the foliations ${\cal F}_{\lambda}$ with a unique invariant ergodic
measure are generic in the space $\Phi_X$ \cite{Mas1}.  In this generic case, 
the dimension group $G_{\lambda}$ can be  identified with a ${\Bbb Z}$-module 
${\Bbb Z}\lambda_1+\dots+{\Bbb Z}\lambda_n$ in the real line ${\Bbb R}$. 









%**************************************************************************
\section{Legendre laminations and their slopes on the modular surface}
%***************************************************************************
Let $\Gamma(N)$ be the principal congruence subgroup of level $N\ge 1$
and $X(N)={\Bbb H}/\Gamma(N)$ the corresponding modular surface. 
The space of geodesic lamination on $X(N)$ is denoted by $\Lambda(X(N))$.
It is known that $\Lambda(X(N))$ is a compact metric space endowed with the
Chabauty topology \cite{CaB}. 


Let $\alpha\in {\Bbb R}-{\Bbb Q}$ be a positive irrational, such that 
the regular continued fraction $\alpha=[a_0, a_1, a_2, \dots]$ has a
property that the matrices $T_i$:
%**************************************************
\begin{eqnarray}\label{eq8}
T_0 &=& \left(\small\matrix{0 & 1\cr 1 & a_0}\right)\cr
T_1 &=& \left(\small\matrix{0 & 1\cr 1 & a_0}\right)\left(\small\matrix{0 & 1\cr 1 & a_1}\right)\cr
T_2 &=& \left(\small\matrix{0 & 1\cr 1 & a_0}\right)\left(\small\matrix{0 & 1\cr 1 & a_1}\right)
        \left(\small\matrix{0 & 1\cr 1 & a_2}\right)\cr
         &\vdots&
\end{eqnarray}
%***************************************************
belong to the group $\Gamma(N)$. Let $\gamma_i$ be an axis of the 
transformation $T_i$ (see appendix). The geodesic lamination $\lambda\in\Lambda(X(N))$ 
is called {\it Legendre} if $\lambda=\lim_{i\to\infty}\gamma_i$ in the Chabauty topology. 
%************************************************************************
\begin{dfn}\label{dfn2}
The positive irrational $\alpha$ is called a slope of the Legendre lamination
$\lambda$ on the modular surface $X(N)$. 
\end{dfn}
%*******************************************************************
Recall that the geodesic laminations $\lambda,\lambda'\Lambda(X(N))$ are
called {\it equivalent} whenever there exists a homeomorphism of the
surface $X(N)$, which sends the (geodesic) leaves of $\lambda$ into
such of $\lambda'$.   
%****************************************************************
\begin{lem}\label{lm1}
The slopes $\alpha,\alpha'$ of the  equivalent
Legendre laminations $\lambda,\lambda'\in\Lambda(X(N))$ 
are the equivalent irrationals $\alpha'=\alpha ~mod~GL(2,{\Bbb Z})$.
\end{lem}
%**************************************************************
{\it Proof.}
Let $\lambda$ be the limit
%**********************************************************
\begin{equation}\label{eq9}
\lambda=\lim_{n\to\infty}T_0 T_1\dots T_n,
\end{equation}
%*******************************************************
and let $\varphi:X(N)\to X(N)$
be a homeomorphism sending $\lambda$ to $\lambda'$. Let us
find a limit formula for $\lambda'$.
First, notice that there exists a bijection between $T_i$,
which occur in the limit of the Legendre laminations, and the
set ${\Bbb Q}^+$ of positive rationals. (Indeed, $T_i$ can be identified with the
partial quotients of the regular continued fractions of irrational numbers.)   
Since $\varphi$ acts on the set of closed geodesics of $X(N)$,
which are the axes of $T_i$, the action of $\varphi$ extends to the positive rationals.
But any such an action has the form:
%************************************************************
\begin{equation}\label{eq10}
\varphi(r)={Ar+B\over Cr+D},\quad r\in {\Bbb Q}^+,\quad
\left(\small\matrix{A & B\cr C & D}\right)\in GL(2,{\Bbb Z}^+).
\end{equation}
%***********************************************************
One can   use the euclidean algorithm to decompose 
$\left(\small\matrix{A & B\cr C & D}\right)$ into the
factors $T_0',\dots, T_k'$, assuming that $k$ is even.
Therefore, the required limit formula for $\lambda'$ has the form:
%**********************************************************
\begin{equation}\label{eq11}
\lambda'=\lim_{n\to\infty}T_0'\dots T_k' T_0 T_1\dots T_n
\end{equation}
%*******************************************************
Since the regular continued fractions of the Legendre laminations 
$\lambda$ and $\lambda'$ have the same tails, we conclude 
that the slopes $\alpha, \alpha'$ are the equivalent irrationals.
$\square$ 







%**************************************************************************
\section{Proof of theorem 1}
%***************************************************************************
(i) Let $\lambda\in\Lambda(X(N))$ be a Legendre lamination and $G_{\lambda}$
the corresponding dimension group. In view \S 4, we will write $G_{\alpha}$,
whenever $\lambda$ has a slope $\alpha$.  Let us prove the following lemma.
%******************************************************
\begin{lem}\label{lm2}
The totally ordered dimension groups $G_{\alpha}$ and $G_{\beta}$ are
order-isomorphic if and only if 
%******************************************************
\begin{equation}\label{eq12}
\beta={a\alpha+b\over c\alpha +d},
\end{equation}
%******************************************************                      
where $a,b,c,d$ are integers such that $ad-bc=\pm 1$. 
\end{lem}
%*****************************************************
{\it Proof.}
Since $G_{\alpha}$ is a dimension group with total order,
the order is generated by a ${\Bbb Z}$-module 
${\Bbb Z}\lambda_1+ \dots  +{\Bbb Z}\lambda_n$ in ${\Bbb R}$,
where 
%************************************************
\begin{equation}\label{eq13}
\lambda_i=\int_{\gamma_i}{\cal F}_{\alpha}d\mu,
\end{equation}
%**********************************************
for a basis $\{\gamma_1,\dots,\gamma_n\}$ in $H_1(X, Sing~{\cal F}_{\alpha}; {\Bbb Z})$.
On the other hand, for the topologically equivalent  foliation ${\cal F}_{\alpha}, {\cal F}_{\beta}$
there exists a basis $\gamma_j'=\sum a_{ij}\gamma_i$, such that
%************************************************
\begin{equation}\label{eq14}
\int_{\gamma_i}{\cal F}_{\beta}d\mu=\int_{\gamma_i'}{\cal F}_{\alpha}d\mu :=\lambda_i',
\end{equation}
%**********************************************
where $(a_{ij})$ is an invertible integer matrix. Using the integration
rules, one obtains:
%************************************************
\begin{equation}\label{eq15}
\lambda_i'=\int_{\sum a_{ij}\gamma_i}{\cal F}_{\alpha}d\mu=
\sum a_{ij}\lambda_i.
\end{equation}
%**********************************************
It is easy to see that the last equation implies that
${\Bbb Z}\lambda_1 +\dots + {\Bbb Z}\lambda_n={\Bbb Z}\lambda_1' +\dots +{\Bbb Z}\lambda_n'$,
i.e. the dimension groups $G_{\alpha}, G_{\beta}$ are order-isomorphic. 
In view of lemma \ref{lm1}, it follows that $\beta=\alpha ~mod ~GL(2, {\Bbb Z})$.  
The inverse is proved similarly.
$\square$


\bigskip\noindent
(ii) Let ${\Bbb Z}\lambda_1+\dots+{\Bbb Z}\lambda_n$ be a ${\Bbb Z}$-module
in ${\Bbb R}$, corresponding to the totally ordered dimension group $G_{\alpha}$.
Consider the Jacobi-Perron continued fraction:
%**********************************************************************************
\begin{equation}\label{eq16}
\left(\matrix{1\cr \theta}\right)=
\lim_{k\to\infty} \left(\matrix{0 & 1\cr I & b_0}\right)\dots
\left(\matrix{0 & 1\cr I & b_k}\right)
\left(\matrix{0\cr {\Bbb I}}\right),
\end{equation}
%************************************************************ 
where $\theta=(\lambda_2/\lambda_1,\dots,\lambda_n/\lambda_1)$ and
$\lambda_1\ne0$. It is known \cite{Bau1}, that the fraction is convergent
on a generic set $V\subset {\Bbb R}^n$ of vectors $\lambda$ corresponding 
to the foliation ${\cal F}_{\alpha}$ with the unique ergodic measure.
In particular, the totally ordered dimension groups are bijective with
such foliations and therefore our fraction is always convergent.

One can now apply a result of Effros and Shen: the dimension group $G_{\alpha}$
is a unimodular dimension group given by the following limit of simplicial dimension 
groups:
%*************************************************************
\begin{equation}\label{eq17}
{\Bbb Z}^n
\buildrel\rm
\left(\matrix{0 & 1\cr I & b_0}\right)
\over\longrightarrow 
{\Bbb Z}^n
\buildrel\rm
\left(\matrix{0 & 1\cr I & b_1}\right)
\over\longrightarrow
{\Bbb Z}^n
\dots
\end{equation}
%**************************************************************  
see \cite{EfSh2}, Corollary 3.3. The item (ii) follows.
$\square$






%**************************************************************************
\section{Appendix}
%***************************************************************************
The present section contains a brief account on the geodesic laminations,
modular surfaces and the Jacobi-Perron continued fractions. The corresponding 
topics are covered in \cite{CaB}, \cite{K} and \cite{B},  respectively. 
%**************************************************************************
\subsection{Geodesics laminations}
%***************************************************************************
Let $S$ be a surface of the constant negative curvature (a hyperbolic surface).
By a {\it geodesic}, one understands  a  maximal arc on $S$ consisting of
the locally shortest sub-arcs (in the given metric on $S$). 
Each geodesic is the image of the open real interval $I$ under a continuous map $I\to S$.
%**************************************************************************
\begin{lem}\label{lm3}
{\bf (Topological classification of geodesics)}
Let $p\in S$ be a point with an attached unit vector $t\in S^1$ on the surface
$S$. Then :

\medskip
(a) for almost all points $t\in S^1$ the geodesic line through $p$ in
direction $t$ is an immersion $I\to S$, i.e. a finite or infinite curve
with the self-intersections;

\smallskip
(b) the complementary set $K\subset S^1$ has the cardinality
of continuum and geodesic lines through $p$ in the direction $t\in K$
are embedded curves of one of the three types:

\medskip\hskip1cm
(i) periodic

\smallskip\hskip1cm
(ii) spiral

\smallskip\hskip1cm
(iii) non-periodic whose closure is a perfect (Cantor) subset of $S$.
\end{lem}
%**************************************************************************
{\it Proof.} See \cite{Art} and \cite{Myr}.
$\square$


\smallskip
The  geodesics of type (b) are called {\it simple}, since they have
no self-crossing or self-tangent points. 
Every geodesic $\gamma: I\to S$ of type (iii) is recurrent, i.e.
for any $t_0\in I$ and $\varepsilon>0$ the $\varepsilon$-neighbourhood
of $p(t_0)$ has infinitely many intersections with $\gamma(t)$ provided
$t>N$, where $N=N(\varepsilon)$ is sufficiently large. 
The topological closure of recurrent geodesic on $S$ contains
 continuum of the disjoint recurrent geodesics,
and called a {\it geodesic lamination} $\lambda$.
The intersection of $\lambda$ with any closed curve on $S$ is a Cantor set. 
The set $S - \lambda$ is called 
a {\it principal region} of $\lambda$. The principal region can have up to 
$4g-4$ connected components on the surface of genus $g\ge2$ \cite{CaB}. 
We denote by $|\lambda|$ the total number of such components. 



A {\it foliation} $\cal F$ on a surface $X$ is partition of $X$ into a disjoint union
of $1$-dimensional and, possibly, a finite number of $0$-dimensional
leaves denoted by $Sing~{\cal F}$. The immediate examples of foliations
are orbits of the flows and trajectories of the quadratic differentials
on $X$.  The foliation $\cal F$ is called {\it measured} if it supports an invariant transversal
measure on the leaves \cite{FLP}. In other words:

\medskip
(i) $Sing~{\cal F}$ consists of the $n$-prong saddles, where $n\ge 3$;

\medskip
(ii) each $1$-leaf is everywhere dense in $X$.


\smallskip\noindent
The geodesic lamination $\lambda$ can be obtained from $\cal F$ by a {\it blow-up}
homotopy. Namely, a {\it separatrix} of $\cal F$ is a $1$-leaf one of whose
ends lie in $Sing~{\cal F}$. The blow-up is a replacement of the separatrix
by a narrow strip $[-\varepsilon,\varepsilon]\times {\Bbb R}$ using a 
homotopy surgery, which does not affect the nearby leaves. The complement
of the blown-up separatrices consists of leaves of $\cal F$ that
make up a perfect (Cantor) set on  $X$. 
It is not hard to see that the above complement
is homeomorphic to $\lambda$.   
Moreover, $|\lambda|$ is equal to the number of singular points
of the foliation ${\cal F}$.  Note that each measured foliations
can be given by the orbits of  a closed one-form $\omega$ on $X$,
passing, if necessary,  to a double cover of $X$ \cite{Mas1}.     







%**************************************************************************
\subsection{Modular surfaces}
%***************************************************************************
Let  $SL(2,{\Bbb Z})$ be the modular group. For every $N=1,2,3,\dots$,
consider a normal subgroup $\Gamma(N)\subset\Gamma$ of finite index consisting
of matrices of the form:
%************************************************************
\begin{equation}\label{eq18}
\left(\matrix{1+Np & Nq\cr Nr & 1+Ns}\right),
\end{equation}
%**********************************************************
where $p,q,r,s\in {\Bbb Z}$. The $\Gamma(N)$ is called
a {\it principal congruence group}  of level $N$.
Let ${\Bbb H}=\{z=x+iy\in {\Bbb C}|y>0\}$ be
the Lobachevsky plane endowed with the hyperbolic metric
$ds=|dz|/ y$. The group $\Gamma(N)$ acts
on ${\Bbb H}$ by the linear-fractional transformations
$z\mapsto {az+b\over cz+d}$, where $\left(\small\matrix{a & b\cr c & d}\right)\in \Gamma(N)$.
The orbit space $X(N)={\Bbb H}/\Gamma(N)$ is an orbifold with a finite number
of cusps corresponding to the  stabilizers of group $\Gamma(N)$.
It follows from the construction that $X(N)$ is a Riemann surface, which is
called  a {\it modular surface}. The genus of $X(N)$
can be evaluated according to the classical formulas \cite{K},
p. 272.


Recall that an element $g\in SL(2,{\Bbb Z})$ is called
(i) elliptic if $|tr~(g)|<2$, (ii) parabolic if $|tr~(g)|=2$, 
or (iii) hyperbolic if $|tr~(g)|>2$.
The cases (i)-(iii) correspond to the transformations
of ${\Bbb H}$ with: (i) one fixed point in $Int~{\Bbb H}$, (ii) one 
fixed point in $\partial {\Bbb H}$, (iii)  two fixed points
in $\partial {\Bbb H}$.  
Let $g$ be hyperbolic, and $C$ the geodesic circle through the fixed
points of $g$. It is known that $g(C)=C$ and $g$  shifts points
of $C$. The hyperbolic length of segment $[x,g(x)]$ does not depend
on $x\in C$ and is calculated according to the formula
$l([x,g(x)])= 2~arc cosh~\left({1\over 2}|tr~(g)|\right)$,
where $tr~(g)$ is the trace of matrix $g\in GL(2,{\Bbb Z})$.
It is not hard to see that the segment $[x,g(x)]$ covers a
periodic geodesic on $X(N)$, whose length coincides with $l([x,g(x)])$.  
The geodesic is called an {\it axis} of the transformation $g$. 




%**************************************************************************
\subsection{Jacobi-Perron fractions}
%***************************************************************************
The Jacobi-Perron algorithm and connected (multidimensional) continued 
fraction generalizes the euclidean algorithm (regular continued fraction)
of an irrational number. Namely, let $\lambda=(\lambda_1,\dots,\lambda_n)$,
$\lambda_i\in {\Bbb R}-{\Bbb Q}$ and  $\theta_{i-1}={\lambda_i\over\lambda_1}$, where
$1\le i\le n$.   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 a {\it  Jacobi-Perron
fraction}. To recover the integers $b_i^{(k)}$ from the vector $(\theta_1,\dots,\theta_{n-1})$,
one has to repeatedly solve the following system of equations:
%*******************************************************************
\begin{equation}
\left\{
\begin{array}{ccc}
\theta_1 &= b_1^{(1)} &+  {1\over\theta_{n-1}'} \nonumber\\
\theta_2 &= b_2^{(1)} &+  {\theta_1'\over\theta_{n-1}'}\nonumber\\
\vdots & \nonumber\\
\theta_{n-1} &= b_{n-1}^{(1)} &+  {\theta_{n-2}'\over\theta_{n-1}'},
\end{array}
\right.
\end{equation}
%*****************************************************************
where $(\theta_1',\dots,\theta_{n-1}')$ is the next input vector.
Thus, each vector $(\theta_1,\dots,\theta_{n-1})$ gives rise to a formal 
Jacobi-Perron continued fraction. Whether the fraction is convergent or not, is yet to be 
determined.  


Let us introduce the following notation. We let $A^{(0)}=\delta_{ij}$
(the Kronecker delta) and
$
A_i^{(k+n)}=\sum_{j=0}^{n-1} b_i^{(k)}A_i^{(\nu +j)}, \quad b_0^{(k)}=1,
$
where $i=0,\dots,n-1$ and $k=0,1,\dots,\infty$. 
The Jacobi-Perron continued fraction  of vector 
$(\theta_1,\dots,\theta_{n-1})$  is said to be {\it convergent}, if 
$\theta_i=\lim_{k\to\infty}{A_i^{(k)}\over A_0^{(k)}}$ 
for all $i=1,\dots,n-1$.
Unless $n=2$, convergence of the Jacobi-Perron fractions is a delicate question. 
To the best of our knowledge, there exists no intrinsic necessary 
and sufficient conditions for such a convergence. However,
the Bauer criterion and the Masur-Veech theorem imply
that the Jacobi-Perron fractions  converge for the generic  vectors $(\theta_1,\dots,\theta_{n-1})$.
Namely, let ${\cal F}$ be a measured foliation
on the surface $X$ of genus $g\ge 1$. Recall that the foliation ${\cal F}$ 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 $V$ in the space of all measured foliations, 
such that each ${\cal F}\in V$ is a uniquely ergodic measured foliation.
We let $\lambda=(\lambda_1,\dots,\lambda_n)$ be the coordinate vector of
the foliation ${\cal F}$. Then the following (Bauer's) criterion 
is true: the Jacobi-Perron continued fraction of $\lambda$  converges 
if and only if $\lambda\in V\subset {\Bbb R}^n$ \cite{Bau1}.




%***********************************************************
\begin{thebibliography}{100}

\bibitem{Art}
E.~Artin, Ein mechanisches System mit quasiergodischen Bahnen,
Abh. Math. Sem. Hamburg (1924), 170-175;
in: Collected papers,  499-504, Springer Verlag. 



\bibitem{Bau1}
M.~Bauer, A characterization of uniquely ergodic interval exchange
maps in terms of the Jacobi-Perron algorithm, Bol. Soc. Bras. Mat. 27 (1996),
109-128. 


\bibitem{B}
L.~Bernstein, The Jacobi-Perron Algorithm, its Theory and Applications,
Lect. Notes in Math. 207, Springer 1971. 



\bibitem{CaB}
A.~J.~Casson and S.~A.~Bleiler, Automorphisms of Surfaces
after Nielsen and Thurston, Lond. Math. Soc.  Student Texts 9, Cambridge, 
1988.


\bibitem{E}    
E.~G.~Effros, Dimensions and $C^*$-Algebras, in: Conf. Board of the Math.
Sciences, Regional conference series in Math., No.46, AMS,  1981.



\bibitem{EfS}
E.~G.~Effros and C.-L.~Shen,
Approximately finite $C^*$-algebras and continued 
fractions, Indiana Univ. Mathematics J. 29 (1980),
191-204. 



\bibitem{EfSh2}
E.~G.~Effros and C.-L.~Shen,
Dimension groups and finite difference equations,
J. Operator Theory 2 (1979), 215-231. 



\bibitem{FLP} 
A.~Fathi,  F.~Laudenbach  and Z.~ Po\'enaru,
Travaux de Thurston sur les surfaces,
~S\'eminaire Orsay, France, Ast\'erisque 66-67 (1979).





\bibitem{G}
K.~R.~Goodearl, Partially Ordered Abelian Groups
with Interpolation, Math. surveys and monographs 20,
AMS 1986.  




\bibitem{K}
A.~W.~Knapp, Elliptic Curves, Princeton, 1992. 


\bibitem{Mas1}
H.~Masur, Interval exchange transformations and measured foliations,
Ann. of Math. (2) 115 (1982), no. 1, 169--200.






\bibitem{Myr}
P.~J.~Myrberg, Ein Approximationssatz f\"ur die Fuchsschen
Gruppen, Acta Math. 57 (1931), 389-409.




\bibitem{Sat}
E.~A.~Sataev, On the number of invariant measures for flows on orientable
surfaces, ~Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no.4, 860-878.


\bibitem{T}
W.~P.~Thurston, The Geometry and Topology of Three-Manifolds,
MSRI 1997, electronic edition of 1980 Princeton Univ. notes,
available at {\sf http://www.msri.org/gt3m/}






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



\vskip1cm

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


\end{document}
math0310400.extracted_bib
\begin{thebibliography}{100}

\bibitem{Art}
E.~Artin, Ein mechanisches System mit quasiergodischen Bahnen,
Abh. Math. Sem. Hamburg (1924), 170-175;
in: Collected papers,  499-504, Springer Verlag. 



\bibitem{Bau1}
M.~Bauer, A characterization of uniquely ergodic interval exchange
maps in terms of the Jacobi-Perron algorithm, Bol. Soc. Bras. Mat. 27 (1996),
109-128. 


\bibitem{B}
L.~Bernstein, The Jacobi-Perron Algorithm, its Theory and Applications,
Lect. Notes in Math. 207, Springer 1971. 



\bibitem{CaB}
A.~J.~Casson and S.~A.~Bleiler, Automorphisms of Surfaces
after Nielsen and Thurston, Lond. Math. Soc.  Student Texts 9, Cambridge, 
1988.


\bibitem{E}    
E.~G.~Effros, Dimensions and $C^*$-Algebras, in: Conf. Board of the Math.
Sciences, Regional conference series in Math., No.46, AMS,  1981.



\bibitem{EfS}
E.~G.~Effros and C.-L.~Shen,
Approximately finite $C^*$-algebras and continued 
fractions, Indiana Univ. Mathematics J. 29 (1980),
191-204. 



\bibitem{EfSh2}
E.~G.~Effros and C.-L.~Shen,
Dimension groups and finite difference equations,
J. Operator Theory 2 (1979), 215-231. 



\bibitem{FLP} 
A.~Fathi,  F.~Laudenbach  and Z.~ Po\'enaru,
Travaux de Thurston sur les surfaces,
~S\'eminaire Orsay, France, Ast\'erisque 66-67 (1979).





\bibitem{G}
K.~R.~Goodearl, Partially Ordered Abelian Groups
with Interpolation, Math. surveys and monographs 20,
AMS 1986.  




\bibitem{K}
A.~W.~Knapp, Elliptic Curves, Princeton, 1992. 


\bibitem{Mas1}
H.~Masur, Interval exchange transformations and measured foliations,
Ann. of Math. (2) 115 (1982), no. 1, 169--200.






\bibitem{Myr}
P.~J.~Myrberg, Ein Approximationssatz f\"ur die Fuchsschen
Gruppen, Acta Math. 57 (1931), 389-409.




\bibitem{Sat}
E.~A.~Sataev, On the number of invariant measures for flows on orientable
surfaces, ~Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no.4, 860-878.


\bibitem{T}
W.~P.~Thurston, The Geometry and Topology of Three-Manifolds,
MSRI 1997, electronic edition of 1980 Princeton Univ. notes,
available at {\sf http://www.msri.org/gt3m/}






\end{thebibliography}