Lattice envelopes

Abstract

We introduce a class of countable groups by some abstract group-theoretic conditions. This class includes linear groups with finite amenable radical and finitely generated residually finite groups with some nonvanishing ${\ell }^2$-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 the possible lattice embeddings of $\Gamma$, that is, of all compactly generated, locally compact groups that contain $\Gamma$ as a lattice. This leads to a precise description of possible nonuniform 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.