The baseleaf preserving mapping class group of the universal hyperbolic solenoid

Abstract:

Given a closed surface $X$, the covering solenoid $\mathbf {X}_\infty$ is by definition the inverse limit of all finite covering surfaces over $X$. If the genus of $X$ is greater than one, then there is only one homeomorphism type of covering solenoid; it is called the universal hyperbolic solenoid. In this paper we describe the topology of $\Gamma (\mathbf {X}_\infty )$, the mapping class group of the universal hyperbolic solenoid. Central to this description is the notion of a virtual automorphism group. The main result is that there is a natural isomorphism of the baseleaf preserving mapping class group of $\mathbf {X}_\infty$ onto the virtual automorphism group of $\pi _1(X,*)$. This may be regarded as a genus independent generalization of the theorem of Dehn, Nielsen, Baer, and Epstein that the pointed mapping class group $\Gamma (X,*)$ is isomorphic to the automorphism group of $\pi _1(X,*)$.