$PSL(2,\mathbb{Z})$ as a non-distorted subgroup of Thompson's group T Ariadna Fossas 20F65Thompson’s group$PSL(2\mathbb{Z})$distortionfree groupfree subgroupsrooted binary treesMinkowski question mark functionpiecewise projective homeomorphisms In this paper we characterize the elements of $PSL_2(\mathbb{Z})$, as a subgroup of Thompson's group $T$, in the language of reduced tree pair diagrams and in terms of piecewise linear maps as well. Actually, we construct the reduced tree pair diagram for every element of $PSL_2(\mathbb{Z})$ in normal form. This allows us to estimate the length of the elements of $PSL_2(\mathbb{Z})$ through the number of carets of their reduced tree pair diagrams and, as a consequence, to prove that $PSL_2(\mathbb{Z})$ is a non-distorted subgroup of $T$. In particular, we find non-distorted free non-abelian subgroups of $T$. Indiana University Mathematics Journal 2011 text pdf 10.1512/iumj.2011.60.4477 10.1512/iumj.2011.60.4477 en Indiana Univ. Math. J. 60 (2011) 1905 - 1926 state-of-the-art mathematics http://iumj.org/access/