Amenable uniformly recurrent subgroups and lattice embeddings
Résumé
We study lattice embeddings for the class of countable groups $\Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $\mathcal{A}_\Gamma$ is continuous. When $\mathcal{A}_\Gamma$ comes from an extremely proximal action and the envelope of $\mathcal{A}_\Gamma$ 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. We then focus on a family of finitely generated groups acting on trees within this class, and show that these embed as cocompact irreducible lattices in some locally compact wreath products. This provides examples of finitely generated simple groups quasi-isometric to a wreath product $C \wr F$, where $C$ is a finite group and $F$ a non-abelian free group.