Teichmüller mapping class group of the universal hyperbolic solenoid

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