Actions of large finite groups on aspherical manifolds

TL;DR

This paper investigates finite group actions on aspherical manifolds, establishing conditions for the Jordan property, symmetry bounds, and rigidity, with applications to locally homogeneous spaces.

Contribution

It provides new results on the structure and symmetry of aspherical manifolds under finite group actions, including conditions for Jordan property and rigidity.

Findings

Homeomorphism group of aspherical manifolds is Jordan under certain conditions

Bounds on the discrete degree of symmetry of aspherical manifolds

Rigidity results for actions on aspherical manifolds

Abstract

In this paper we study actions of finite groups on closed connected aspherical manifolds. Under some assumptions on the outer automorphism group of the fundamental group of a closed connected aspherical manifold $M$, we prove that the homeomorphism group of $M$ is Jordan, we bound the discrete degree of symmetry of $M$ and obtain a rigidity result, and we study the number of stabilizers of finite group actions on $M$. Thereafter, we show that closed connected aspherical locally homogeneous spaces $H\setminus G/\Gamma$ satisfy the necessary hypothesis on the outer automorphism group of the fundamental group.