On subgroups of minimal topological groups

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.

To appear in Topology and its Applications. 39 pages

References in corpus (5)

• Topological groups: where to from here?
• Compactifications of topological groups
• Globalization of the partial isometries of metric spaces and local approximation of the group of isometries of the Urysohn space
• Representations of residually finite groups by isometries of the Urysohn space
• Some isometry groups of Urysohn space

Cited by in corpus (18)

• Topological groups: where to from here?
• Globalization of Confluent Partial Actions on Topological and Metric Spaces
• Ambitable topological groups
• Algebraically determined topologies on permutation groups
• Homomorphism and embedding universal structures for restricted classes
• Automatic continuity for isometry groups
• Representations of residually finite groups by isometries of the Urysohn space
• Equivariant concentration in topological groups
• New algebras of functions on topological groups arising from G-spaces
• On unitary representations of groups of isometries
• Each topological group embeds into a duoseparable topological group
• Ramsey-Milman phenomenon, Urysohn metric spaces, and extremely amenable groups
• Isometry groups among topological groups
• The Urysohn sphere is oscillation stable
• Generalizing separability, precompactness and narrowness in topological groups
• The isometry group of the Urysohn space as a Levy group
• Equivariant dissipation in non-archimedean groups
• Group topologies on automorphism groups of homogeneous structures