Thompson's group $T$ was one of the first examples of finitely
presented infinite simple groups. There are at least three
different ways of representing the elements of Thompson's group
$T$ (see \cite{CFP} for a detailed introduction to Thompson's
groups). Probably the most common interpretation is as a subgroup
of the group of homeomorphisms of the circle, thought as the unit
interval with identified endpoints. Then, $T$ is the group of
orientation preserving piecewise linear homeomorphisms of the
circle that are differentiable except at finitely many dyadic
rational numbers, and such that, on intervals of
differentiability, the derivatives are powers of 2 (see, for
example, \cite{sergiescuGhys}).

Another interesting approach to Thompson's group $T$ describes it
as equivalence classes of tree pair diagrams. A tree pair diagram
is a pair of finite rooted binary trees with the same number of
leaves, with a cyclic numbering system pairing the leaves in the
two trees. We say that two different tree pair diagrams are
equivalent if they have the same reduced representant. This
combinatorial version of Thompson's group $T$ allowed Burillo,
Cleary, Stein and Taback to discuss metric properties of $T$ (see
\cite{BCST}). In particular, they found an estimation of the word
length in terms of the number of carets in a reduced tree pair
diagram, and showed that some subgroups of $T$ are undistorted.

We need to mention a third viewpoint about Thompson's group $T$.
We can represent $T$ as a subgroup of the group of piecewise
projective orientation preserving homeomorphisms of the real
projective line $\mathcal{R}P^1$. This result was claimed
separately by Thurston (see \cite{CFP}) and Kontsevich, and then
proved by Imbert (see \cite{imbert}) and Sergiescu (see
\cite{sergiescu}). It turns out that $T$ is isomorphic to
$PPSL_2(\mathbb{Z})$, which is the group of orientation preserving
homeomorphisms of the real projective line which are piecewise
$PSL_2(\mathbb{Z})$, and have a finite numbers of non
differentiable points, all of them being rational numbers. This
geometric approach is also related to the fact that one can
actually see Thompson's group $T$ as the asymptotic mapping class
group of an infinite surface of genus zero (see
\cite{funarKapoudjian}).

The piecewise projective approach to Thompson's group $T$ invites
us to study the projective special linear group
$PSL_2(\mathbb{Z})$ as a subgroup of $T$ using the combinatorial
methods of tree pair diagrams. It turns out that all the reduced
tree pair diagrams representing elements of $PSL_2(\mathbb{Z})$ as
a subgroup of Thompson's group $T$ have exactly one leaf and one
interior vertex for each level. In fact, we provide an explicit
bijection between tree pair diagrams of elements in
$PSL_2(\mathbb{Z})$ and the corresponding reduced words on the
classical generators of $PSL_2(\mathbb{Z})$ of order 2 and 3 (see
\cite{humphreys}). This characterization of $PSL_2(\mathbb{Z})$ in
terms of tree pair diagrams together with the results of Burillo,
Cleary, Stein and Taback in the word metric on $T$ (see
\cite{BCST}) allow us to prove that $PSL_2(\mathbb{Z})$ is a non
distorted subgroup of Thompson's group $T$.

Although it was already known that $T$ has subgroups isomorphic to
the free non abelian group of rank 2 (this is the usual way to
proof the non amenability of $T$), it was not known if these
subgroups are distorted. The piecewise projective approach
together with our results give an easy example of a non distorted
subgroup of $T$ isomorphic to $F_2$. There are several normal
subgroups of $PSL_2(\mathbb{Z})$ isomorphic to the free non
abelian group of rank two. One particular example is the
commutator group of $PSL_2(\mathbb{Z})$ (see \cite{humphreys}). In
addition, this subgroup has finite index on $PSL_2(\mathbb{Z})$.
Now, using the fact that finite index subgroups are non distorted
and our result that $PSL_2(\mathbb{Z})$ is non distorted as a
subgroup of Thompson's group $T$, we have constructed a non
distorted subgroup of Thompson's group $T$ isomorphic to the free
non abelian group of rank 2.


\subsection{Definitions and statements of the results}

First we state the definitions related to the three different ways
of representing Thompson's group $T$ introduced above:

\begin{definition}
The {\rm{piecewise linear Thompson's group $T$}} is the set of
orientation preserving piecewise linear homeomorphisms of the
circle $S^1=[0, 1]/_{0\sim1}$ that are differentiable except at
finitely many dyadic rational numbers and such that, on intervals
of differentiability, the derivatives are powers of 2.
\end{definition}

For the combinatorial definition we need the following notion:

\begin{definition} A {\rm{tree pair diagram}} is a triple
$(T_1,\sigma,T_2)$, where $T_1$ and $T_2$ are finite rooted binary
trees with the same number $n$ of leaves, and $\sigma$ is a cyclic
permutation of the set $\{1,\ldots, n\}$. The binary tree $T_1$ is
called the {\rm{source tree}}, and $T_2$ is called the {\rm{target
tree}}. A node of an ordered rooted binary tree together with its
two downward directed edges is called a {\rm{caret}}. A caret is
called {\rm{exposed}} if it contains two leaves of the tree. A
tree pair diagram is {\rm{reduced}} if, for all exposed carets of
$T_1$, the images under $\sigma$ of their leaves do not form an
exposed caret of $T_2$.
\end{definition}

When elements of Thompson's group $T$ are represented as
equivalence classes of tree pair diagrams, the multiplication of
$(T_1,\sigma_1, T_2)$ by $(T_3,\sigma_2,T_4)$ is described by the
following procedure.
\begin{enumerate}
\item Let $T_{23}$ be an ordered rooted binary tree which is a
common expansion of $T_2$ and $T_3$. This element always exists.
\item Let $(T_1',\sigma_1',T_{23})$ and $(T_{23},\sigma_2',T_4')$
be the tree pair diagrams which are equivalent to $(T_1,\sigma_1,
T_2)$ and $(T_3,\sigma_2,T_4)$, respectively. \item Then, one sets
$(T_1,\sigma_1, T_2) * (T_3,\sigma_2,T_4) =(T_1',
\sigma_1'\sigma_2', T_4')$, and reduces the obtained tree pair
diagram, if possible.
\end{enumerate}

\begin{definition}
  The {\rm{combinatorial Thompson's group $T$}} is the set of
  equivalence classes of tree pair diagrams.
\end{definition}

The third viewpoint of Thompson's group $T$ connects it with the
projective special linear group $PSL_2(\mathbb{Z})$ (see, for
example, \cite{martin}).

\begin{definition} The {\rm{piecewise projective Thompson's group $T$}}
($PPSL_2(\mathbb{Z})$) is the group of orientation preserving
homeomorphisms of the real projective line $\mathbb{R}P^1$ which
are piecewise $PSL_2(\mathbb{Z})$ and have a finite number of non
differentiable points, all of them being rational numbers.
\end{definition}

Now we need a few properties of the projective special linear
group. The group $PSL_2(\mathbb{Z})$ is isomorphic to the free
product $\mathbb{Z}/2\mathbb{Z}\ast\mathbb{Z}/3\mathbb{Z}$, which
means that it admits a generating set $\{a,b\}$, where $a^2=1$ and
$b^3=1$ (see \cite{arbres}, page 20 example 1.5.3). Then, every
element of $PSL_2(\mathbb{Z})$ has a normal form in the generators
$a$ and $b$ given by a word of the form
$a^{\epsilon_1}b^{\delta_1}ab^{\delta_2}a \ldots
ab^{\delta_k}a^{\epsilon_2}$, where $k$ is a non negative integer,
$\epsilon_1,\epsilon_2 \in \{0,1\}$ and
$\delta_1,\ldots,\delta_k\in \{-1,1\}$. We will denote by $a$ and
$b$ both the generators of $PSL_2(\mathbb{Z})$ as a group of
matrices and their images in $T$ under the isomorphism between
$PPSL_2(\mathbb{Z})$ and Thompson's group $T$.

In this paper we give a characterization of the elements of
$PSL_2(\mathbb{Z}) \subset T$ in terms of reduced tree pair
diagrams. Before stating the theorem we need the following
definition:

\begin{definition} A finite rooted binary tree is called {\rm{thin}} if
all its carets have one leave and one internal vertex except for
the last caret, which is exposed.
\end{definition}

\remark It is easy to see that one can characterize thin trees
with $n$ leaves by associating a weight $r_i \in \{-1,1\}$ to each
one of its $n-2$ internal vertices. Let $v_0, \ldots, v_{n-3}$ be
the internal vertices of a thin tree. Denote by $v_{-1}$ its root.
Then, $r_i=1$ if $v_i$ is the left descendant of $v_{i-1}$ and
$r_i=-1$ if $v_i$ is the right descendant of $v_{i-1}$.

The main result of this note is the following:

\begin{theorem}
  Let $(T_1,\sigma,T_2)$ be the reduced tree pair diagram of an element $f$ of Thompson's group $T$.
  Then, $f$ belongs to the subgroup $PSL_2(\mathbb{Z})$ if and only if $(T_1,\sigma,T_2)$
  satisfies one of the following conditions:
  \begin{enumerate}
    \item the number of leaves of $T_1$ and $T_2$ is less than 4; or
    \item the trees $T_1$ and $T_2$ are thin with associated weights $r_0, \ldots, r_{k-1}$ and $s_0, \ldots,
    s_{k-1}$ respectively, which verify the equations:
    \beq \sum_{i=2}^{k-1}r_i s_{k+1-i}= 2 - k, \label{eq1} \enq
    and
    \beq l + \sigma(1) + \epsilon(s_0) \equiv  \frac{3 - s_1}{2} \quad
    ({\rm{mod}} \, k+2),\label{eq2}
    \enq
    where $l-1$ is the cardinal of the set $\{i \, : \, r_i=-1, 0 \leq i \leq k-1\}$, and
    $\epsilon(x) = \left\{ \begin{array}{cl}
    1, & \text{\rm{if} } x = 1,\\
    0, & \text{\rm{if} } x = -1.
    \end{array} \right.$
  \end{enumerate} \label{mainThm}
\end{theorem}

This answers a question posed by Vlad Sergiescu.

In fact, given an element of $PSL_2(\mathbb{Z})$ as a subgroup of
Thompson's group $T$ in its normal form on the generating set
$\{a,b\}$, we describe an easy way to construct its reduced tree
pair diagram.

\begin{proposition}
  Let $w(a,b) = a^{\epsilon_1}b^{\delta_1}a b^{\delta_2}a \ldots a b^{\delta_k}a^{\epsilon_2}$
  be a reduced word in the standard generators $\{a,b\}$ of $PSL_2(\mathbb{Z})$, viewed as a subgroup of Thompson's group
  $T$, i.e. $\epsilon_1,\epsilon_2 \in \{0,1\}$ and $\delta_1,\ldots,\delta_k\in \{-1,1\}$. Assume that $k \geq 2$.
  Let $T_1$ be the thin tree given by the weights $r_0 = \epsilon^{-1}(\epsilon_1)$ and $r_i = \delta_i$ for $1 \leq
  i \leq k-1$, and let $T_2$ be the thin tree given by the weights $s_0 = \epsilon^{-1}(\epsilon_2)$ and $s_i = -\delta_{k+1-i}$ for
  $1 \leq i \leq k-1$. Let $\sigma$ be the cyclic permutation
  defined by
  \beq \sigma(1) \equiv \frac{3-s_1}{2} - \epsilon(s_0) - l \quad ({\rm{mod}} \, k+2),\label{eq3} \enq
  where  $l$ and $\epsilon$ are defined as in theorem 1.
  Then, $(T_1, \sigma, T_2)$ is the reduced tree pair diagram for $w$.\\
  Furthermore, the weights $r_0, \ldots, r_{k-1}$ and $s_0,\ldots, s_{k-1}$
  satisfy the equations of theorem \ref{mainThm}. \label{treesWord}
\end{proposition}

This proposition proves the `if' part of the main theorem. For the
`only if' part it suffices to show that the number of solutions to
the equations (\ref{eq1}) and (\ref{eq2}) coincides with the
number of words $w(a,b) = a^{\epsilon_1}b^{\delta_1}a
b^{\delta_2}a \ldots a b^{\delta_k}a^{\epsilon_2}$, with
$\epsilon_1,\epsilon_2 \in \{0,1\}$ and
$\delta_1,\ldots,\delta_k\in \{-1,1\}$.

Now, we can ask whether or not $PSL_2(\mathbb{Z})$ is distorted as
a subgroup of Thompson's group $T$. Recall that a finitely
generated subgroup $H$ of a finitely generated group $G$ is
distorted if there exists an infinite family $\{h_n\}_{n \in
\mathbb{N}}$ of elements of $H$ such that
\begin{enumerate}
 \item $\displaystyle |h_n|_Z < |h_{n+1}|_Z$,
 \item $\displaystyle \lim_{n \rightarrow \infty} |h_n|_Z = \infty$, and
 \item the limit
 $ \lim_{n \rightarrow \infty} \frac{|h_n|_Z}{|h_n|_Y}$
 exists and is equal to zero,
\end{enumerate}
where $Z$ is a finite generating set for the group $G$ which
contains a finite generating set $Y$ for the subgroup $H$, and
$|.|_Z$ and $|.|_Y$ denote the word metrics on the generating sets
$Z$ and $Y$, respectively. Observe that $|.|_Z \leq |.|_Y$. See
\cite{grAsympInv} for another definition of distortion.

\remark $H$ is a non distorted subgroup of $G$ if and only if
there are constants $K > 0$ and $L$ such that
$$ \frac{1}{K}|h|_Y - L \leq |h|_Z \leq K|h|_Y + L$$
holds for every $h \in H$. \label{dist}

Using theorem \ref{mainThm} and proposition \ref{treesWord} we can
estimate the length of the elements in this subgroup. Furthermore,
Burillo, Cleary, Stein and Taback gave a general estimation of the
length of elements in $T$ in the number of carets in \cite{BCST}.
As a consequence of both results, we obtain:

\begin{proposition}
  The group $PSL_2(\mathbb{Z})$ is a non distorted subgroup of
  Thompson's group $T$. \label{noDist}
\end{proposition}

In particular, we derive:

\begin{corollary}
  Let $\{a,b\}$ be the standard generating set of $PSL_2(\mathbb{Z})$ as a subgroup of Thompson's group $T$.
  Then, the subgroup $H=\left<abab,a\bar{b}a\bar{b}\right>$ is a free non abelian group of rank 2
  and it is non distorted in $T$.
\end{corollary}

Finally, we consider the elements of $PSL_2(\mathbb{Z})$ as
piecewise linear maps and we give a characterization in terms of
the coordinates of their non differentiable points $(x_0,y_0)$,
$(x_1,y_1)$, $\ldots$, $(x_k,y_k)$. As the result is technical and
needs both notation and definitions, the reader is referred to
section 4 for the details.

\textbf{Structure of the paper.} This paper is structured in four
sections. The group $T$ is presented as the group of piecewise
$PSL_2(\mathbb{Z})$ homeomorphisms of the real projective line in
section two. The third section deals with the characterization of
the elements of $PSL_2(\mathbb{Z})$ as a subgroup of $T$, the
proof that $PSL_2(\mathbb{Z})$ is a non distorted subgroup and the
construction of a non distorted free group of rank two in $T$.
Finally, in section four we give a piecewise linear
characterization of the elements of $PSL_2(\mathbb{Z})$.


\subsection{Acknowledgements.}

The author wishes to thank Jos\'e Burillo, Louis Funar and Vlad
Sergiescu for their comments and useful discussions. This work was
supported by ``La Caixa'' and ``l'Ambassade de France en Espagne''
Postgraduated Fellowship.
{"config": "arxiv", "file": "1006.0508/1Intro.tex"}