Logic Seminar

A subset of a Polish group is Roelcke precompact if, given any open subset $ V $ of the group, it can be covered by finitely many sets of the form $ V f V $. Such sets form an ideal and the familiar Roelcke precompact groups are those for which this ideal is improper. A group is said to be locally Roelcke precompact when this ideal countains an open set. Examples of such groups---in addition to all Roelcke precompact groups and all locally compact groups---include the isometry group of the Urysohn metric space and the automorphism group of the countably-regular tree.

All locally Roelcke precompact groups are locally bounded in the sense of the coarse geometry of topological groups developed by C. Rosendal. Indeed, we characterize them as those locally bounded Polish groups for which every coarsely bounded subset is Roelcke precompact. We also characterize them as those groups whose completions with respect to their Roelcke (or lower) uniformities are locally compact. We also assess the conditions under which this locally compact space carries the structure of a semi-topological semigroup extending multiplication in the group.