Non-abelian group structure on the Urysohn space

TL;DR

This paper demonstrates the existence of a non-abelian group structure on the Urysohn space by constructing a free group with a compatible metric, expanding understanding of algebraic structures on this universal metric space.

Contribution

It introduces a novel variant of the Graev metric to construct a non-abelian group structure on the Urysohn space, which was previously unknown.

Findings

Existence of a non-abelian group structure on the Urysohn space

Construction of a free group with a two-sided invariant metric

The metric is isometric to the rational Urysohn space

Abstract

Following the continuing interest in the Urysohn space and, more specifically, the recent problem area of finding and comparing group structures on the Urysohn space we prove that there exists a non-abelian group structure on the Urysohn universal metric space. More precisely, we introduce a variant of the Graev metric that enables us to construct a free group with countably many generators equipped with a two-sided invariant metric that is isometric to the rational Urysohn space. We provide several open questions and problems related to this subject.