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arXiv · 2104.07111

Quantifying local embeddings into finite groups

Henry Bradford
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Abstract

We study a function $\mathcal{L}_Γ$ which quantifies the LEF (local embeddability into finite groups) property for a finitely generated group $Γ$. 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 $\mathcal{L}_Γ$ 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.

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Henry Bradford. 2021-08-06. Quantifying local embeddings into finite groups. https://arxiv.org/abs/2104.07111

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