Galois cohomology and profinitely solitary Chevalley groups (Q6624794)

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-connected absolutely almost simple split linear \(k\)-group. Then \(G\) satisfies the finite splitting principle if and only if \N\begin{itemize}\N\item[(i)] \(k\) is totally imaginary and \(G\) is arbitrary, or \N\item[(ii)] \(k\) has precisely one real place and \(G\) has type \(\mathsf{A}_{2n+1}\) or \(\mathsf{C}_{n}\), or \N\item[(iii)] \(k\) is arbitrary and \(G\) has type \(\mathsf{A}_{2n}\).\N\end{itemize}