Amenable uniformly recurrent subgroups and lattice embeddings

TL;DR

This paper investigates how certain countable groups with a specific amenable subgroup property can be embedded as lattices in larger locally compact groups, revealing structural restrictions based on their amenable subgroups.

Contribution

It introduces new conditions under which countable groups with amenable uniformly recurrent subgroups can be embedded as lattices, especially when associated with extremely proximal actions.

Findings

Restrictions on normal subgroups of G

Limitations on product decompositions of G

Conditions for dense mappings to product groups

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.