Teichmüller mapping class group of the universal hyperbolic solenoid

Abstract

We show that the homotopy class of a quasiconformal self-map of the universal hyperbolic solenoid H_∞ is the same as its isotopy class and that the uniform convergence of quasiconformal self-maps of H_∞ to the identity forces them to be homotopic to conformal maps. We identify a dense subset of T(H_∞) such that the orbit under the base leaf preserving mapping class group MCG_(BLP)(H_∞) of any point in this subset has accumulation points in the Teichmüller space T(H_∞). Moreover, we show that finite subgroups of MCG_(BLP)(H_∞) are necessarily cyclic and that each point of T(H_∞) has an infinite isotropy subgroup in MCG_(BLP)(H_∞).

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