The group of isometries of a locally compact metric space with one end

TL;DR

This paper investigates the dynamics of isometry groups acting on locally compact metric spaces with one end, revealing finitely many pseudo-components and conditions for proper action.

Contribution

It introduces the concept of pseudo-components in this context and characterizes the properness of the isometry group's action based on the space's structure.

Findings

X has finitely many pseudo-components

Exactly one pseudo-component is non-compact

G acts properly on the space

Abstract

In this note we study the dynamics of the natural evaluation action of the group of isometries $G$ of a locally compact metric space $(X,d)$ with one end. Using the notion of pseudo-components introduced by S. Gao and A. S. Kechris we show that $X$ has only finitely many pseudo-components of which exactly one is not compact and $G$ acts properly on. The complement of the non-compact component is a compact subset of $X$ and $G$ may fail to act properly on it.