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

TL;DR

The paper demonstrates that finite index subgroups of PSL_2(Z) can be conjugated by homeomorphisms that are nearly conformal but not conformal, revealing complex dynamics in the Teichmüller space of the punctured solenoid.

Contribution

It establishes the existence of near-conformal conjugations between finite index subgroups of PSL_2(Z) that are not conformal, impacting the understanding of Teichmüller space dynamics.

Findings

Existence of non-conformal conjugations close to conformal for finite index subgroups.

The orbit closure of the basepoint in Teichmüller space is uncountably large.

The orbit closure strictly contains the orbit itself.

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.