The Teichm\"uller distance between finite index subgroups of $PSL_2(\mathbb{Z})$

Abstract

For a given $\epsilon >0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+\epsilon)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $\epsilon>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+\epsilon)$-quasiconformal between them that is not homotopic to a conformal map. As an application of the above results, we show that the orbit of the basepoint in the Teichm\"uller space $T(\S)$ of the punctured solenoid $\S$ under the action of the corresponding Modular group (which is the mapping class group of $\S$ \cite{NS}, \cite{Odd}) has the closure in $T(\S)$ strictly larger than the orbit and that the closure is necessarily uncountable.