Free Groups in Lattices

TL;DR

This paper proves that finitely generated free groups can be embedded into lattices of certain locally compact groups, showing a strong structural relationship under specific conditions.

Contribution

It establishes that free groups are virtually subgroups of lattices in a broad class of locally compact groups, extending known embedding results.

Findings

Finitely generated free groups are virtually subgroups of lattices in certain groups.

Embedding requires small perturbations and passage to finite index subgroups.

Additional hypotheses are needed when the quotient space is noncompact, including for G=SO(n,1).

Abstract

Let G be any locally compact, unimodular, metrizable group. The main result of this paper, roughly stated, is that if F<G is any finitely generated free group and \Gamma < G any lattice, then up to a small perturbation and passing to a finite index subgroup, F is a subgroup of \Gamma. If G/\Gamma is noncompact then we require additional hypotheses that include G=SO(n,1).