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

TL;DR

This paper explores the structure of finite subgroups of homeomorphism groups of locally symmetric manifolds, showing the diversity of possible subgroup configurations and characterizing rigidity properties of lattice actions.

Contribution

It demonstrates that for any finite group, there exist locally symmetric manifolds with that group as the maximal finite subgroup, and classifies the possible numbers of conjugacy classes of finite subgroups.

Findings

Existence of manifolds with any prescribed finite group as G(M)

The number of conjugacy classes of finite subgroups can be one, countably many, or continuum

Complete characterization of topological local and strong rigidity for lattice actions

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.