Coarse Geometry and P. A. Smith Theory

TL;DR

This paper introduces a generalized fixed point concept called the bounded fixed set for group actions on metric spaces and proves an analogue of P. A. Smith theorem relating coarse homology of these sets.

Contribution

It extends fixed point theory to coarse geometry by defining the bounded fixed set and establishing a Smith-type theorem for spaces of finite asymptotic dimension.

Findings

Established the bounded fixed set as a generalization of fixed points.

Proved an analogue of P. A. Smith theorem in the coarse geometric setting.

Linked the coarse homology of bounded fixed sets to that of the entire space.

Abstract

We define a generalization of the fixed point set, called the bounded fixed set, for a group acting by isometries on a metric space. An analogue of the P. A. Smith theorem is proved for metric spaces of finite asymptotic dimension, which relates the coarse homology of the bounded fixed set to the coarse homology of the total space.