Some universal constructions in abstract topological dynamics

TL;DR

This paper surveys universal constructions in topological dynamics, introduces a new intrinsic characterization of extremely amenable groups, and discusses related problems, examples, and open questions in the field.

Contribution

It provides an intrinsic description of extremely amenable groups, solving a longstanding problem from 1967, and reviews key constructions and results in the area.

Findings

Intrinsic description of extremely amenable groups

Negative solution to Teleman's 1957 conjecture

Discussion of examples and open questions in topological dynamics

Abstract

This small survey of basic universal constructions related to the actions of topological groups on compacta is centred around a new result --- an intrinsic description of extremely amenable topological groups (i.e., those having a fixed point in each compactum they act upon), solving a 1967 problem by Granirer. Another old problem whose solution (in the negative) is noted here, is a 1957 conjecture by Teleman on irreducible representations of general topological groups. Our exposition covers the greatest ambit and the universal minimal flow, as well as closely related constructions and results from representation theory. We present in a simplified fashion the known examples of extremely amenable groups and discuss their relationship with a 1969 problem by Ellis. Open questions, including those from bordering disciplines, are reviewed along the way.