Generic norms and metrics on countable abelian groups

TL;DR

This paper explores the generic properties of invariant metrics on countable abelian groups, showing that for unbounded groups, typical metrics lead to spaces isometric to the Urysohn space, with some metrics resulting in extremely amenable completions.

Contribution

It generalizes previous results by demonstrating the prevalence of Urysohn space isometries and extreme amenability in the space of invariant metrics on certain countable abelian groups.

Findings

Dense set of metrics makes the group isometric to the rational Urysohn space.

Comeager set of metrics yields completions isometric to the Urysohn space.

For certain groups, all metrics in a comeager set produce the same metric group after completion.

Abstract

For a countable abelian group $G$ we investigate generic properties of the space of all invariant metrics on $G$. We prove that for every such an unbounded group $G$, i.e. group which has elements of arbitrarily high order, there is a dense set of invariant metrics on $G$ which make $G$ isometric to the rational Urysohn space, and a comeager set of invariant metrics such that the completion is isometric to the Urysohn space. This generalizes results of Cameron and Vershik, Niemiec, and the author. Then we prove that for every $G$ such that $G\cong \bigoplus_\mathbb{N} G$ there is a comeager set of invariant metrics on $G$ such that all of them give rise to the same metric group after completion. If moreover $G$ is unbounded, then using a result of Melleray and Tsankov we get that the completion is extremely amenable.