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.

v1: 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