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Non-abelian group structure on the Urysohn universal space

Michal Doucha (2015)

Fundamenta Mathematicae

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 list several related open problems.

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