Metric topological groups: their metric approximation and metric ultraproducts

TL;DR

This paper introduces a method to approximate separable topological groups with left-invariant metrics using metric ultraproducts of finite groups, connecting to concepts like sofic and hyperlinear groups.

Contribution

It constructs a metric ultraproduct framework that embeds all separable topological groups with left-invariant metrics, advancing the understanding of metric approximations.

Findings

Existence of a countable sequence of finite groups approximating any separable topological group

Metric ultraproducts contain isometric copies of all separable groups with left-invariant metrics

Connections established with sofic, hyperlinear, and weakly sofic groups

Abstract

We define a metric ultraproduct of topological groups with left-invariant metric, and show that there is a countable sequence of finite groups with left-invariant metric whose metric ultraproduct contains isometrically as a subgroup every separable topological group with left-invariant metric. In particular, there is a countable sequence of finite groups with left-invariant metric such that every finite subset of an arbitrary topological group with left-invariant metric may be approximated by all but finitely many of them. We compare our results with related concepts such as sofic groups, hyperlinear groups and weakly sofic groups.