Controlling LEF growth in some group extensions

TL;DR

This paper investigates the LEF growth function of finitely generated groups, showing that a wide class of functions can be approximated by such groups' growth, using constructions involving semidirect products and subgroup estimates.

Contribution

It introduces methods to realize a broad range of growth functions as LEF growth of finitely generated groups through semidirect product constructions and subgroup analysis.

Findings

Any sufficiently smooth increasing function between n! and exp(exp(n)) can be approximated by a LEF group's growth.

Estimates on LEF growth are obtained for groups formed as semidirect products involving permutation groups and automorphism groups.

Identification of sequences of finitely presented subgroups with short relative presentations is a key tool in the analysis.

Abstract

We study the LEF growth function of a finitely generated LEF group $\Gamma$, which measures the orders of finite groups admitting local embeddings of balls in a word metric on $\Gamma$. We prove that any sufficiently smooth increasing function between $n!$ and $\exp(\exp(n))$ is close to the LEF growth function of some finitely generated group. This is achieved by estimating the LEF growth of some semidirect products of the form $FSym (\Omega) \rtimes \Gamma$, where $\Gamma \curvearrowright \Omega$ is an appropriate transitive action, and $FSym (\Omega)$ is the group of finitely supported permutations of $\Omega$. A key tool in the proof is to identify sequences of finitely presented subgroups with short "relative" presentations. In a similar vein we also obtain estimates on the LEF growth of some groups of the form $E_{\Omega} (R) \rtimes \Gamma$, for $R$ an appropriate unital ring and $E_{\Omega} (R)$ the subgroup of $Aut_R (R[\Omega])$ generated by all transvections with respect to basis $\Omega$.