Quantifying local embeddings into finite groups

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.

22 pages, 1 figure. Version 2 adds a new section on metric approximations, and other improvements to the exposition and minor corrections