Teichmuller mapping class group of the universal hyperbolic solenoid

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