On universal minimal proximal flows of topological groups

TL;DR

This paper investigates the effectiveness of actions of certain topological groups on their universal minimal proximal flows, providing conditions for effectiveness and extending group actions via commensurators.

Contribution

It establishes criteria for the effectiveness of group actions on universal minimal proximal flows and extends Furstenberg's theorem to broader contexts.

Findings

Effectiveness of group actions on flows depends on group properties.

Necessary and sufficient conditions for effectiveness are identified.

Extension of actions via commensurators is demonstrated.

Abstract

In this paper, we show that the action of a characteristically simple, non-extremely amenable (non-strongly amenable, non-amenable) group on its universal minimal (minimal proximal, minimal strongly proximal) flow is effective. We present necessary and sufficient conditions, for the action of a topological group with trivial center on its universal minimal proximal flow, to be effective. A theorem of Furstenberg about the isomorphism of the universal minimal proximal flows of a discrete group and its subgroups of finite index ([Theorem~II.4.4]) is strengthened. Finally, for a pair of groups $H < G$ the same method is applied in order to extend the action of $H$ on its universal minimal proximal flow to an action of its commensurator group $\mathrm{Comm}_G(H)$.