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 $\dim M\ne 4$) 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$M\ne 4$) of proper discontinuous actions of uniform lattices in semisimple Lie groups on the associated symmetric spaces.