On the action of the group of isometries on a locally compact metric space

TL;DR

This paper investigates the convergence behavior of nets of isometries on locally compact metric spaces, establishing conditions under which subsequences converge to isometries on pseudo-components, and provides simplified proofs of classical theorems.

Contribution

It characterizes the convergence of nets of isometries in locally compact metric spaces and offers concise proofs of established theorems.

Findings

Existence of a subnet converging to an isometry on a pseudo-component.

Convergence of nets implies convergence on pseudo-components.

Simplified proofs of van Dantzig--van der Waerden and Gao--Kechris theorems.

Abstract

In this short note we give an answer to the following question. Let $X$ be a locally compact metric space with group of isometries $G$. Let $\{g_i\}$ be a net in $G$ for which $g_ix$ converges to $y$, for some $x,y\in X$. What can we say about the convergence of $\{g_i\}$? We show that there exist a subnet $\{g_j\}$ of $\{g_i\}$ and an isometry $f:C_x\to X$ such that $g_{j}$ converges to $f$ pointwise on $C_x$ and $f(C_x)=C_{f(x)}$, where $C_x$ and $C_y$ denote the pseudo-components of $x$ and $y$ respectively. Applying this we give short proofs of the van Dantzig--van der Waerden theorem (1928) and Gao--Kechris theorem (2003).