The baseleaf preserving mapping class group of the universal hyperbolic solenoid

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 $\pi_1(X,*)$.