On subgroups of minimal topological groups |
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Authors: | Vladimir V Uspenskij |
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Institution: | Department of Mathematics, 321 Morton Hall, Ohio University, Athens, OH 45701, USA |
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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 U1 is the Urysohn universal metric space of diameter 1, the group Iso(U1) of all self-isometries of U1 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. |
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Keywords: | primary 22A05 secondary 06F05 22A15 54D35 54E15 54E50 54H11 54H12 54H15 57S05 |
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