Conditional Non-Soficity of p-adic Deligne Extensions: on a Theorem of Gohla and Thom
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| Format: | Preprint |
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2024
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| _version_ | 1866909429627092992 |
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| author | Chapman, Michael Dikstein, Yotam Lubotzky, Alexander |
| author_facet | Chapman, Michael Dikstein, Yotam Lubotzky, Alexander |
| contents | 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 $\widetildeΓ$ of arithmetic lattices $Γ$ 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 $\widetildeΓ$ 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 $Γ$ is positive, generalizing [Evra, Kaufman, 2016]. This would imply that $Γ$ is stable.
(b) To prove that the (flexible) stability of $Γ$ implies the non-soficity of $\widetildeΓ$.
Clause (b) was proved by Gohla and Thom. Here we offer a more algebraic/combinatorial treatment to their theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_02913 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditional Non-Soficity of p-adic Deligne Extensions: on a Theorem of Gohla and Thom Chapman, Michael Dikstein, Yotam Lubotzky, Alexander Combinatorics Group Theory 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 $\widetildeΓ$ of arithmetic lattices $Γ$ 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 $\widetildeΓ$ 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 $Γ$ is positive, generalizing [Evra, Kaufman, 2016]. This would imply that $Γ$ is stable. (b) To prove that the (flexible) stability of $Γ$ implies the non-soficity of $\widetildeΓ$. Clause (b) was proved by Gohla and Thom. Here we offer a more algebraic/combinatorial treatment to their theorem. |
| title | Conditional Non-Soficity of p-adic Deligne Extensions: on a Theorem of Gohla and Thom |
| topic | Combinatorics Group Theory |
| url | https://arxiv.org/abs/2410.02913 |