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

Abstract

For a given $ε>0$, we show that there exist two finite index subgroups of $PSL_2(\mathbb{Z})$ which are $(1+ε)$-quasisymmetrically conjugated and the conjugation homeomorphism is not conformal. This implies that for any $ε>0$ there are two finite regular covers of the Modular once punctured torus $T_0$ (or just the Modular torus) and a $(1+ε)$-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 Teichmüller space $T(§)$ of the punctured solenoid $§$ under the action of the corresponding Modular group (which is the mapping class group of $§$ \cite{NS}, \cite{Odd}) has the closure in $T(§)$ strictly larger than the orbit and that the closure is necessarily uncountable.

23 pages, 5 Figures