Amenable uniformly recurrent subgroups and lattice embeddings

Abstract

We study lattice embeddings for the class of countable groups $\Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $A_{\Gamma }$ is continuous. When $A_{\Gamma }$ comes from an extremely proximal action and the envelope of $A_{\Gamma }$ is co-amenable in $\Gamma$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $\Gamma$ 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.

arXiv commentv1: 44 pages, preliminary version. v2: slightly modified version. v3: modified terminology, added paragraph 6.5.4. v4: Part of Section 6 has been extracted to arXiv:2001.08689

References in corpus (11)

• Simplicity and superrigidity of twin building lattices
• C*-simplicity and the unique trace property for discrete groups
• Decomposing locally compact groups into simple pieces
• Isometry groups of non-positively curved spaces: discrete subgroups
• Normal amenable subgroups of the automorphism group of the full shift
• Uniformly recurrent subgroups and the ideal structure of reduced crossed products
• Finitely presented groups associated with expanding maps
• A lattice in more than two Kac--Moody groups is arithmetic
• Rigidity properties of full groups of pseudogroups over the Cantor set
• New simple lattices in products of trees and their projections
• Approximating simple locally compact groups by their dense locally compact subgroups

Cited by in corpus (1)

• Locally compact wreath products