Lattice envelopes

TL;DR

This paper defines a new class of countable groups with specific properties and characterizes their possible lattice embeddings within locally compact groups, providing insights into their structure and applications to geometric group theory.

Contribution

It introduces a broad class of groups and precisely describes their lattice embeddings, including non-uniform cases and applications to negatively curved manifolds.

Findings

Characterization of lattice embeddings for the new class of groups

Description of non-uniform lattice embeddings

Applications to fundamental groups of negatively curved manifolds

Abstract

We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group $\Gamma$ in this class we determine the general structure of its possible lattice embeddings, i.e. of all compactly generated, locally compact groups that contain $\Gamma$ 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.