Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions

Abstract

To any compact hyperbolic Riemann surface $X$, we associate a new type of automorphism group -- called its *commensurability automorphism group*, $ComAut(X)$. The members of $ComAut(X)$ arise from closed circuits, starting and ending at $X$, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by $X$ (in $T_{\infty}$), for the action of the universal commensurability modular group on the universal direct limit of Teichmüller spaces, $T_{\infty}$. Now, each point of $T_{\infty}$ represents a complex structure on the universal hyperbolic solenoid. We notice that $ComAut(X)$ acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing $X$ is *arithmetic*. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over $T_{\infty}$, are studied by us.