Galois cohomology and profinitely solitary Chevalley groups
type: Item
instance of: scholarly article
title: Galois cohomology and profinitely solitary Chevalley groups
author: Holger Kammeyer
author: Ryan Spitler
published in: Mathematische Annalen
publication date: 2024-10-28T00:00:00Z
review text: Let \(\Gamma\) be a finitely generated residually finite group and \(\widehat{\Gamma}\) its profinite completion. Then \(\Gamma\) is (absolutely) profinitely solitary if every other finitely generated residually finite group \(\Delta\) with \(\widehat{\Delta} \approx \widehat{\Gamma}\) satisfies \(\Delta \approx \Gamma\) (where \(\Delta \approx \Gamma\) if and only if \(\Delta\) and \(\Gamma\) are abstractly commensurable).\N\NThe aim of the paper under review is to examine which arithmetic subgroups of split simple algebraic groups (Chevalley groups) are profinitely solitary. Let \(k\) be a number field and denote the ring of finite adeles of \(k\) by \(\mathbb{A}_{k}^{f}\). The field \(k\) is locally determined if every number field \(l\) with \(\mathbb{A}_{l}^{f} \simeq \mathbb{A}_{k}^{f}\) satisfies \(l \simeq k\). The main results of the paper are the following.\N\NTheorem 1: Let \(k\) be a locally determined number field and let \(G\) be a simply-connected, absolutely almost simple split linear \(k\)-group such that \N\begin{itemize}\N\item[(i)] \(k\) is totally imaginary and \(G\) is not of type \(\mathsf{A}_{1}\) if \(k\) is imaginary quadratic; or \N\item[(ii)] \(k\) has precisely one real place and either \(G\) has type \(\mathsf{A}_{2n+1}\) with \(n \geq 0\) (\(n \geq 1\) for \(k=\mathbb{Q}\)) or \(G\) has type \(\mathsf{C}_{n}\) with \(n \geq 2\), or \N\item[(iii)] \(k\) is arbitrary and \(G\) has type \(\mathsf{A}_{2n}\) with \(n \geq 1\), or \N\item[(iv)] \(k=\mathbb{Q}\) and \(G\) has type \(\mathsf{B}_{3}\), \(\mathsf{B}_{4}\), \(\mathsf{D}_{4}\), \(\mathsf{D}_{5}\), or \(\mathsf{G}_{2}\).\N\end{itemize} \NThen either all arithmetic subgroups of \(G\) are profinitely solitary, or there exists an arithmetic subgroup of \(G\) that has a proper Grothendieck subgroup.\N\NA subgroup \(\Delta\) of \(\Gamma\) is a Grothendieck subgroup if the inclusion induces an isomorphism \(\widehat{\Delta}\simeq \widehat{\Gamma}\).\N\NTheorem 2: Let \(G\) be a simply-co…
reviewed by: Enrico Jabara
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 11E72
zbMATH DE Number: 7932362
zbMATH Keywords: profinite group
zbMATH Keywords: profinite completion
zbMATH Keywords: Chevalley group
zbMATH Keywords: cohomology
MaRDI profile type: Publication
cites work: Galois and Cartan cohomology of real groups
cites work: The Bianchi groups are separable on geometrically finite subgroups.
cites work: Profinite completions and Kazhdan's property (T).
cites work: Hyperplane sections in arithmetic hyperbolic manifolds
cites work: Q3999894
cites work: Absolute profinite rigidity and hyperbolic geometry
cites work: Adelic superrigidity and profinitely solitary lattices
cites work: On absolutely profinitely solitary lattices in higher rank Lie groups
cites work: On the profinite rigidity of lattices in higher rank Lie groups
cites work: Q4389329
cites work: On the adele rings and zeta-functions of algebraic number fields
cites work: Subgroup separability and virtual retractions of groups
cites work: Some more Non-arithmetic Rigid groups
cites work: Q3997989
cites work: Q4002684
cites work: On the existence of isotropic forms of semi-simple algebraic groups over number fields with prescribed local behavior.
cites work: On the congruence subgroup problem
cites work: On the congruence subgroup problem. II
cites work: Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres.
cites work: Q2756583
cites work: The monogeneity of radical extensions
cites work: Linear algebraic groups.
DOI: 10.1007/S00208-024-02808-W
description: scientific article; zbMATH DE number 7932362