# Controlling LEF growth in some group extensions

## TL;DR

This paper explores how certain mathematical structures called LEF groups grow, by analyzing their properties and constructing specific examples.

## Contribution

The paper proves that a wide range of growth functions can be realized by finitely generated LEF groups through semidirect products.

## Key findings

- Any sufficiently smooth increasing function between n! and exp(exp(n)) can approximate the LEF growth of some finitely generated group.
- The LEF growth of semidirect products of the form FSym(Ω) ⋊ Γ is estimated using transitive actions and finitely supported permutations.
- Estimates for LEF growth are also obtained for groups of the form EΩ(R) ⋊ Γ using ring-based constructions.

## Abstract

For a finitely generated LEF group \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ, we study the orders of finite groups admitting local embeddings of balls in a word metric on \documentclass[12pt]{minimal}
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\begin{document}$$\Gamma $$\end{document}Γ, as measured by the LEF growth function. We prove that any sufficiently smooth increasing function between n! and \documentclass[12pt]{minimal}
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\begin{document}$$\exp (\exp (n))$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{FSym}\,}}(\Omega ) \rtimes \Gamma $$\end{document}FSym(Ω)⋊Γ, where \documentclass[12pt]{minimal}
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\begin{document}$$\Omega \curvearrowleft \Gamma $$\end{document}Ω↶Γ is an appropriate transitive action and \documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{FSym}\,}}(\Omega )$$\end{document}FSym(Ω) is the group of finitely supported permutations of \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$E_{\Omega } (R) \rtimes \Gamma $$\end{document}EΩ(R)⋊Γ, for R an appropriate unital ring and \documentclass[12pt]{minimal}
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\begin{document}$$E_{\Omega } (R)$$\end{document}EΩ(R) the subgroup of \documentclass[12pt]{minimal}
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\begin{document}$${{\,\textrm{Aut}\,}}_R (R[\Omega ])$$\end{document}AutR(R[Ω]) generated by all transvections with respect to basis \documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document}Ω.