Quantifying local embeddings into finite groups

TL;DR

This paper introduces a new function to measure how finitely generated groups can be locally embedded into finite groups, compares it with residual finiteness, and explores its properties and implications in group theory.

Contribution

It defines the LEF growth function, computes it for specific groups, compares it with residual finiteness and soficity, and proves the non-computability of universal bounds for non-cyclic groups.

Findings

LEF growth function computed for certain wreath products

Existence of residually finite groups with many local embeddings but few finite quotients

Universal upper bound on LEF growth functions is non-computable

Abstract

We study a function $\mathcal{L}_{\Gamma}$ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group $\Gamma$. We compute this "LEF growth" function in some examples, including certain wreath products. We compare LEF growth with the analogous quantitative version of residual finiteness, and exhibit a family of finitely generated residually finite groups which nevertheless admit many more local embeddings into finite groups than they do finite quotients. Along the way, we give a new proof that B.H. Neumann's continuous family of $2$-generated groups contains no finitely presented group, a result originally due to Baumslag and Miller. We compare $\mathcal{L}_{\Gamma}$ with quantitative versions of soficity and other metric approximation properties of groups. Finally, we show that there exists a "universal" function which is an upper bound on the LEF growth of any group on a given number of generators, and that (for non-cyclic groups) any such function is non-computable.