A trichotomy theorem for transformation groups of locally symmetric manifolds and topological rigidity

Abstract

Let $M$ be a locally symmetric irreducible closed manifold of dimension $\ge 3$. A result of Borel [Bo] combined with Mostow rigidity imply that there exists a finite group $G = G(M)$ such that any finite subgroup of $\text{Homeo}^+(M)$ is isomorphic to a subgroup of $G$. Borel [Bo] asked if there exist $M$'s with $G(M)$ trivial and if the number of conjugacy classes of finite subgroups of $\text{Homeo}^+(M)$ is finite. We answer both questions: (1) For every finite group $G$ there exist $M$'s with $G(M) = G$, and (2) the number of maximal subgroups of $\text{Homeo}^+(M)$ can be either one, countably many or continuum and we determine (at least for $\dim M \neq 4$) when each case occurs. Our detailed analysis of (2) also gives a complete characterization of the topological local rigidity and topological strong rigidity (for dim$M\neq 4$) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.