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Amenable uniformly recurrent subgroups and lattice embeddings

Part of: Topological dynamics Lie groups Locally compact groups and their algebras Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  07 February 2020

ADRIEN LE BOUDEC
ADRIEN LE BOUDEC*
Affiliation:
UCLouvain, IRMP, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve, Belgium CNRS, Unité de Mathématiques Pures et Appliquées, ENS-Lyon, France email [email protected]
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Abstract

We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in $\unicode[STIX]{x1D6E4}$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.

Keywords

lattices in locally compact groupsstrongly proximal actionsChabauty spaceuniformly recurrent subgroupsgroups acting on trees

MSC classification

Primary: 20E08: Groups acting on trees 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 22D05: General properties and structure of locally compact groups 22E40: Discrete subgroups of Lie groups
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 5 , May 2021 , pp. 1464 - 1501
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Copyright
© The Author(s) 2020. Published by Cambridge University Press

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