On subgroups of minimal topological groups

By V. Uspenskij. 2000

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Abstract A topological group is minimal if it does not admit a strictly coarser Hausdorff group topology. The Roelcke uniformity (or lower uniformity) on a topological group is the greatest lower bound of the left and right uniformities. A group is Roelcke-precompact if it is precompact with respect to the Roelcke uniformity. Many naturally arising non-Abelian topological groups are Roelcke-precompact and hence have a natural compactification. We use such compactifications to prove that some groups of isometries are minimal. In particular, if U 1 is the Urysohn universal metric space of diameter 1, the group Iso ( U 1 ) of all self-isometries of U 1 is Roelcke-precompact, topologically simple and minimal. We also show that every topological group is a subgroup of a minimal topologically simple Roelcke-precompact group of the form Iso ( M ) , where M is an appropriate non-separable version of the Urysohn space.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cfd4846b339b00e13196253f016b58f65845c166