On the structure and arithmeticity of lattice envelopes

TL;DR

This paper investigates the structure and arithmetic properties of lattice embeddings across a broad class of countable groups, including linear groups, hyperbolic groups, and convergence groups.

Contribution

It provides new results on the structure and arithmeticity of lattice envelopes for diverse classes of countable groups.

Findings

Results on the structure of lattice embeddings

Conditions for arithmeticity of lattice envelopes

Applicability to various classes of groups such as hyperbolic and linear groups

Abstract

We announce results about the structure and arithmeticity of all possible lattice embeddings of a class of countable groups which encompasses all linear groups with simple Zariski closure, all groups with non-vanishing first l2-Betti number, word hyperbolic groups, and, more general, convergence groups.