The Teichmüller distance between finite index subgroups of PSL <inf>2</inf>(ℤ)

For a given ε > 0, we show that there exist two finite index subgroups of PSL2(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 T0 (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(Sp) of the punctured solenoid Sp under the action of the corresponding Modular group (which is the mapping class group of Sp [6], [7]) has the closure in T(Sp) strictly larger than the orbit and that the closure is necessarily uncountable. © 2008 Springer Science+Business Media B.V.