Jordan property for homeomorphism groups and almost fixed point property

TL;DR

This paper investigates properties of finite group actions on topological manifolds, establishing bounds and fixed point properties that hold after passing to subgroups of bounded index, with results applicable to various classes of manifolds.

Contribution

It introduces bounds on symmetry groups and fixed point properties for finite group actions on manifolds, extending previous results to broader classes of topological manifolds.

Findings

Existence of a uniform bound on subgroup index with fixed points

Results apply to manifolds with finitely generated homology

Establishment of the Jordan property for certain manifolds

Abstract

We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These include the Jordan property, the almost fixed point property, as well as bounds on the discrete symmetry group. Most of our results apply to manifolds satisfying some restriction such as having nonzero Euler characteristic or having the integral homology of a sphere. For an arbitrary topological manifold $X$ such that $H_*(X;{\mathbf Z})$ is finitely generated, we prove the existence a constant $C$ with the property that for any continuous action of a finite group $G$ on $X$ such that every $g\in G$ fixes at least on point of $X$, there is a subgroup $H\leq G$ satisfying $[G:H]\leq C$ and a point $x\in X$ which is fixed by all elements of $H$.