Controlling LEF growth in some group extensions

Abstract For a finitely generated LEF group $$\Gamma $$ Γ , we study the orders of finite groups admitting local embeddings of balls in a word metric on $$\Gamma $$ Γ , as measured by the LEF growth function . We prove that any sufficiently smooth increasing function between n ! and $$\exp (\exp (n))$$ 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 $${{\,\textrm{FSym}\,}}(\Omega ) \rtimes \Gamma $$ FSym ( Ω ) ⋊ Γ , where $$\Omega \curvearrowleft \Gamma $$ Ω ↶ Γ is an appropriate transitive action and $${{\,\textrm{FSym}\,}}(\Omega )$$ FSym ( Ω ) is the group of finitely supported permutations of $$\Omega $$ 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 $$E_{\Omega } (R) \rtimes \Gamma $$ E Ω ( R ) ⋊ Γ , for R an appropriate unital ring and $$E_{\Omega } (R)$$ E Ω ( R ) the subgroup of $${{\,\textrm{Aut}\,}}_R (R[\Omega ])$$ Aut R ( R [ Ω ] ) generated by all transvections with respect to basis $$\Omega $$ Ω .