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. 615–636DOI 10.4171/GGD/450