Vladimir V. Uspenskij

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.

A compact group which is not Valdivia compact

A compact space K is Valdivia compact if it can be embedded in a Tikhonov cube I A in such a way that the intersection K fl Σ is dense in K, where Σ is the sigma-product (= the set of points with countably many non-zero coordinates). We show that there exists a compact connected Abelian group of weight ω 1 which is not Valdivia compact, and deduce that Valdivia compact spaces are not preserved by open maps.

The Roelcke compactification of groups of homeomorphisms

Abstract Let X be a zero-dimensional compact space such that all non-empty clopen subsets of X are homeomorphic to each other, and let Aut X be the group of all self-homeomorphisms of X , equipped with the compact-open topology. We prove that the Roelcke compactification of Aut X can be identified with the semigroup of all closed relations on X whose domain and range are equal to X . We use this to prove that the group Aut X is topologically simple and minimal.

A note on a question of R. Pol concerning light maps

Abstract Let f :X→Y be an onto map between compact spaces such that all point-inverses of f are zero-dimensional. Let A be the set of all functions u :X→I=[0,1] such that u[f ← (y)] is zero-dimensional for all y∈Y . Do almost all maps u :X→I , in the sense of Baire category, belong to A ? Torunczyk proved that the answer is yes if Y is countable-dimensional. We extend this result to the case when Y has property C .

EFFECTIVE MINIMAL SUBFLOWS OF BERNOULLI FLOWS

We show that every innite discrete group G has an innite minimal subow in its Bernoulli ow f0; 1g G. A countably innite group G has an eective minimal subow in f0; 1g G. If G is countable and residually nite then it has such a subow which is free. We do not know whether there are groups G with no free subows in f0; 1g G.