# Metric topological groups: their metric approximation and metric ultraproducts

### Michal Doucha

Czech Academy of Sciences, Prague, Czechia, and University of Franche-Comté, Besançon, France

## 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.

## Cite this article

Michal Doucha, Metric topological groups: their metric approximation and metric ultraproducts. Groups Geom. Dyn. 12 (2018), no. 2, pp. 615–636

DOI 10.4171/GGD/450