TY - JOUR

T1 - On universal minimal proximal flows of topological groups

AU - Dai, Xiongping

AU - Glasner, Eli

N1 - Publisher Copyright:
© 2018 American Mathematical Society.

PY - 2019/3

Y1 - 2019/3

N2 - 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 ([Proximal flows, Springer-Verlag, Berlin-New York, 1976]) 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 CommG(H).

AB - 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 ([Proximal flows, Springer-Verlag, Berlin-New York, 1976]) 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 CommG(H).

KW - Effective action

KW - Strong/extreme amenability

KW - Universal minimal proximal flow

UR - http://www.scopus.com/inward/record.url?scp=85065057530&partnerID=8YFLogxK

U2 - 10.1090/proc/14292

DO - 10.1090/proc/14292

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AN - SCOPUS:85065057530

SN - 0002-9939

VL - 147

SP - 1149

EP - 1164

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 3

ER -