On the isometrization of groups of homeomorphisms

TL;DR

This paper establishes criteria under which groups of homeomorphisms can be endowed with invariant metrics, linking isometrizability to properties like equiregularity and properness, with significant theorems for various topological spaces.

Contribution

It introduces the concepts of equiregularity and nearly proper actions and proves theorems characterizing when a group of homeomorphisms is (properly) isometrizable based on these properties.

Findings

Isometrizability characterized by equiregularity for certain spaces.

Proper isometrizability requires equiregularity and nearly properness.

Proper actions imply proper isometrizability for locally compact spaces.

Abstract

Let $G$ be a group of homeomorphisms of a topological space $X$. $G$ is $\textit{(properly) isometrizable}$ if there exists a $G$-invariant (proper) gauge structure on $X$. $G$ is $\textit{equiregular}$ if for every $x \in X$ and every open neighborhood $U$ of $x$ in $X$ there is an open neighborhood $V$ of $x$ in $X$ such that $cl(V) \subset U$ and every $y \in X$ has an open neighborhood $N_y$ with the property that for every $g \in G$, if $g(N_y) \cap cl(V) \neq \emptyset$, then $g(N_y) \subset U$. $G$ is $\textit{nearly proper}$ if for all compact subsets $A$ and $B$ of $X$, $cl$ ( $\bigcup$ { $g(A):g\in G$ and $g(A)\cap B \neq \emptyset$ } ) is compact. $G$ $\textit{acts properly on}$ $X$ if for all compact subsets $A$ and $B$ of $X$, the subset $G_{A,B}$ = { $g\in G : g(A) \cap B \neq \emptyset$ } is compact when $G$ is endowed with the compact-open topology. THE ISOMETRIZATION THEOREM: If $X$ is a Hausdorff space and $G$ \ $X$ is a paracompact regular space, then: $G$ is isometrizable if and only if $G$ is equiregular. THE PROPER ISOMETRIZATION THEOREM: If $X$ is a locally compact $\sigma$-compact Hausdorff space and $G$ \ $X$ is a regular space, then: $G$ is properly isometrizable if and only if $G$ is equiregular and nearly proper. The PROPER ISOMETRIZATION THEOREM has the following corollary. THEOREM OF ABEL-MANOUSSOS-NOSKOV: If $X$ is a locally compact $\sigma$-compact Hausdorff space and $G$ acts properly on $X$, then $X$ is properly isometrizable.