Lattice envelopes

Abstract

We introduce a class of countable groups by some abstract grouptheoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing ℓ 2 -Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group Γ in this class we determine the general structure of the possible lattice embeddings of Γ, i.e. of all compactly generated, locally compact groups that contain Γ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.

Model geometries of finitely generated groups

“…When these conditions are satisfied, they obtain results that are stronger and more complete than the results of this article. The results of this article complement [BFS20] by providing information on finitely generated groups in which at least one of the conditions (Irr), (CAF) and (NbC) does not hold. Both the statement and proof of Proposition 4.12 of this article are similar to and inspired by Proposition 5.3 of [BFS20].…”

“…The results of this article complement [BFS20] by providing information on finitely generated groups in which at least one of the conditions (Irr), (CAF) and (NbC) does not hold. Both the statement and proof of Proposition 4.12 of this article are similar to and inspired by Proposition 5.3 of [BFS20]. However, we replace the (CAF) condition, which says the group does not contain an infinite amenable commensurated subgroup, with the condition that the group does not contain an infinite finite rank abelian commensurated subgroup.…”

“…Mosher-Sageev-Whyte studied model geometries of virtually free groups [MSW03, MSW02]. Furman and Bader-Furman-Sauer classified lattice envelopes for a large class of groups, including lattices in Lie groups and S-arithmetic lattices [Fur01,BFS20]. Dymarz studied model geometries of certain solvable groups [Dym15].…”

Super-rigidity and non-linearity for lattices in products

“…We refer the reader to [Mon05] where such a theorem is proven, along with a very nice converse: under certain natural assumptions on a finitely generated lattice in a product, the only way it could have a linear representation with an infinite image is if it is an arithmetic lattice to begin with. For a similar conclusion (in characteristic ) with no finite generation assumption, see [BFS19, Theorem 4.5].…”

Graphically discrete groups and rigidity

“…The lattice envelope problem has roots in Mostow's Strong Rigidity [Mos73], which classifies the Lie groups that contain the fundamental group of a closed hyperbolic manifold as a uniform lattice. This classification was extended to lattice embeddings into locally compact groups and for a wider class of finitely generated groups by Furman [Fur01] Date: March 10, 2023. and Bader-Furman-Sauer [BFS20]. Dymarz [Dym15] classified lattice envelopes of certain solvable groups, and the classification of lattice envelopes of virtually free groups was given by Mosher-Sageev-Whyte [MSW02].…”