On relative extreme amenability

TL;DR

This paper explores the concept of relative extreme amenability in topological groups, providing characterizations, introducing new concepts like interpolants, and applying the theory to dynamical systems and Fra"{i}ssé structures.

Contribution

It introduces the notion of an extremely amenable interpolant and extends the theory of relative extreme amenability with new characterizations and applications.

Findings

Relative extreme amenability characterized by fixed point properties

Existence of examples of maximally relatively extremely amenable pairs

New conditions for universal minimal spaces of automorphism groups

Abstract

The purpose of this paper is to study the notion of relative extreme amenability for pairs of topological groups. We give a characterization by a fixed point property on universal spaces. In addition we introduce the concepts of an extremely amenable interpolant as well as maximally relatively extremely amenable pairs and give examples. It is shown that relative extreme amenability does not imply the existence of an extremely amenable interpolant. The theory is applied to generalize results of Kechris, Pestov and Todorcevic relating to the application of Fra\"{i}ss\'e theory to the theory of Dynamical Systems. In particular, new conditions enabling to characterize universal minimal spaces of automorphism groups of Fra\"{i}ss\'e structures are given.