Controlling LEF growth in some group extensions

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 $FSym (Ω) \rtimes Γ$, where $Γ\curvearrowright Ω$ 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 $E_Ω (R) \rtimes Γ$, for $R$ an appropriate unital ring and $E_Ω (R)$ the subgroup of $Aut_R (R[Ω])$ generated by all transvections with respect to basis $Ω$.

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