The Teichmüller distance between finite index subgroups of PSL_2ℤ

Abstract

For a given ϵ>0, we show that there exist two finite index subgroups of PSL_2(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 map 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(S^p) of the punctured solenoid S^p under the action of the corresponding Modular group (which is the mapping class group of S^p [6], [7]) has the closure in T(S^p) strictly larger than the orbit and that the closure is necessarily uncountable.

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