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Furstenberg boundaries for pairs of groups

Part of: Noncompact transformation groups Connections with other structures, applications

Published online by Cambridge University Press:  20 February 2020

NICOLAS MONOD Open the ORCID record for NICOLAS MONOD [Opens in a new window]
NICOLAS MONOD*
Affiliation:
EPFL, CH-1015Lausanne, Switzerland email nicolas.monod@epfl.ch
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Abstract

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$. If we consider in addition a subgroup $H<G$, the relative notion of $(G,H)$-boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$. In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$, namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$. We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

Keywords

group actionstopological dynamicsamenability

MSC classification

Primary: 54H20: Topological dynamics
Secondary: 22F05: General theory of group and pseudogroup actions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1514 - 1529
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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