Conditional Non-Soficity of p-adic Deligne Extensions: on a Theorem of Gohla and Thom

TL;DR

This paper investigates the non-soficity of certain group extensions related to p-adic arithmetic lattices, providing algebraic and combinatorial methods to extend previous results on their non-approximability by almost-homomorphisms.

Contribution

It offers a new algebraic and combinatorial proof that certain group extensions are non-sofic, building on and extending prior results in the area.

Findings

Established non-soficity of specific p-adic lattice extensions.

Provided algebraic/combinatorial proof of Gohla and Thom's theorem.

Connected stability properties to non-soficity of group extensions.

Abstract

A long standing problem asks whether every group is sofic, i.e., can be separated by almost-homomorphisms to the symmetric group $Sym(n)$. Similar problems have been asked with respect to almost-homomorphisms to the unitary group $U(n)$, equipped with various norms. One of these problems has been solved for the first time in [De Chiffre, Gelbsky, Lubotzky, Thom, 2020]: some central extensions $\widetilde{\Gamma}$ of arithmetic lattices $\Gamma$ of $Sp(2g,\mathbb{Q}_p)$ were shown to be non-Frobenius approximated by almost homomorphisms to $U(n)$. Right after, it was shown that similar results hold with respect to the $p$-Schatten norms in [Lubotzky, Oppenheim, 2020]. It is natural, and has already been suggested in [Chapman, Lubotzky, 2024] and [Gohla, Thom, 2024], to check whether the $\widetilde{\Gamma}$ are also non-sofic. In order to show that they are (also) non-sofic, it suffices: (a) To prove that the permutation Cheeger constant of the simplicial complex underlying $\Gamma$ is positive, generalizing [Evra, Kaufman, 2016]. This would imply that $\Gamma$ is stable. (b) To prove that the (flexible) stability of $\Gamma$ implies the non-soficity of $\widetilde{\Gamma}$. Clause (b) was proved by Gohla and Thom. Here we offer a more algebraic/combinatorial treatment to their theorem.