Equivariant concentration in topological groups

Abstract

We prove that, if G is a second-countable topological group with a compatible right-invariant metric d and (µn) n∈N is a sequence of compactly supported Borel probability measures on G converging to invariance with respect to the mass transportation distance over d and such that (spt µn, d↾spt µn , µn ↾spt µn ) n∈N concentrates to a fully supported, compact mm-space (X, dX, µX ), then X is homeomorphic to a G-invariant subspace of the Samuel compactification of G. In particular, this confirms a conjecture by Pestov and generalizes a well-known result by Gromov and Milman on the extreme amenability of topological groups. Furthermore, we exhibit a connection between the average orbit diameter of a metrizable flow of an arbitrary amenable topological group and the limit of Gromov's observable diameters along any net of Borel probability measures UEB-converging to invariance over the group.

Big Ramsey degrees and topological dynamics

“…We do this by first developing the theory of G-equicontinuous ultracoproducts, then defining weak containment via the property that a flow weakly contains another iff an ultracopower of the former factors onto the latter. The ultracoproduct construction for compact Hausdorff spaces was thoroughly developed by Bankston in a series of works [3,4,5], and the ultracoproduct construction of G-flows is implicit in work of Schneider [36] connecting topological dynamics to Gromov's metric measure geometry. When the group G is allowed to be any topological group, a major difficulty of working at this level of generality is that the theory of weak containment becomes extremely subtle.…”