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Following the continuing interest in the Urysohn space and, morespecifically, the recent problem area of finding and comparing group structureson the Urysohn space we prove that there exists a non-abelian group structureon the Urysohn universal metric space. More precisely, we introduce a variantof the Graev metric that enables us to construct a free group with countablymany generators equipped with a two-sided invariant metric that is isometric tothe rational Urysohn space. We provide several open questions and problemsrelated to this subject.

Non-abelian group structure on the Urysohn space