Controlling LEF growth in some group extensions

Abstract

We study the LEF growth function of a finitely generated LEF group Γ, which measures the orders of finite groups admitting local embeddings of balls in a word metric on Γ. 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(Ω) ⋊ Γ, where ΓΩ is an appropriate transitive action, and FSym(Ω) is the group of finitely supported permutations of Ω. 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 EΩ(R) ⋊ Γ, for R an appropriate unital ring and EΩ(R) the subgroup of AutR(R[Ω]) generated by all transvections with respect to basis Ω.