Topological combinatorics and amenability
Final Report Abstract
Within this project, we developed new approximation techniques for amenable topological groups by means of finite combinatorial objects, e.g., bipartite graphs. Among the main results of the project is a characterization of amenability for topological groups in terms of almost invariant finite subsets, a topological analogue of Følner’s classical amenability criterion for discrete groups. This result gives rise to an invariant averaging technique that has interesting consequences for the coarse geometry and ergodic theory of amenable topological groups. Furthermore, it allows one to approximate an amenable topological group by amenable group actions on sets, rather than topological spaces. A central concept that appeared during the project is the so-called UEB topology on the group algebra, i.e., the topology of uniform convergence on uniformly equicontinuous, bounded subsets of the algebra of uniformly continuous, bounded functions on a topological group. The study of this topology and the corresponding notion of asymptotic invariance turned out to be crucial for many aspects of the project. Apart from the above, this topology was used to study the extreme amenability of topological groups of measurable maps and to prove new amenability criteria for certain classes of topological groups in terms of fixed points for continuous affine actions on convex compact subsets of different types of locally convex spaces, e.g., Hilbert spaces.
Publications
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On Følner sets in topological groups. 34 pages, August 2016
F. M. Schneider, A. B. Thom
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On amenability and groups of measurable maps. 12 pages, January 2017
V. G. Pestov, F. M. Schneider
