Cohomological goodness and the profinite completion of Bianchi groups.
type: Item
instance of: scholarly article
title: Cohomological goodness and the profinite completion of Bianchi groups.
zbMATH Open document ID: 1194.20029
DOI: 10.1215/00127094-2008-031
published in: Duke Mathematical Journal
publication date: 2008-08-05T00:00:00Z
review text: In this accessible and carefully written paper, the authors explore the concept of cohomological `goodness', which was introduced by \textit{J.-P. Serre} [Galois cohomology. Springer Monographs in Mathematics, Berlin: Springer (2002; Zbl 1004.12003)] to describe the situation where the cohomology of a group \(G\) is strongly linked to the cohomology of its profinite completion \(\widehat G\). It is proved that various types of groups are `good'; among these are Bianchi groups and limit groups. The paper also contains examples of groups which are not good -- arithmetic subgroups of semisimple groups with the congruence subgroup property -- and an application of cohomological goodness to central extensions of Fuchsian groups. A group \(G\), with profinite completion \(\widehat G\), is said to be `good' if for every finite \(G\)-module \(M\) and all natural numbers \(n\) the homomorphism \(H^n(\widehat G,M)\to H^n(G,M)\), induced by the natural homomorphism \(G\to\widehat G\), is an isomorphism between cohomology groups. For instance, one can show that finitely generated free groups and surface groups are good. One of the main results of the paper under review is that Bianchi groups are good. Bianchi groups are arithmetic Kleinian groups of the form \(\text{PSL}(2,\mathcal O_d)\), where \(\mathcal O_d\) is the ring of integers of the imaginary quadratic number field \(\mathbb{Q}(\sqrt{-d})\) for a squarefree positive integer \(d\). Moreover, every non-cocompact arithmetic subgroup of \(\text{PGL}_2(\mathbb{C})\) is commensurable to a Bianchi group, and one may conjecture that all Kleinian groups are good. Another main result of the paper is that limit groups are good. Limit groups are the finitely generated fully residually free groups; they play a prominent role in Sela's solution of Tarski's problem about equivalence of first order theories of finitely generated non-Abelian free groups. The proofs of these main results are based on the fact that Bianchi groups as wel…
reviewed by: Benjamin Klopsch
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 11F75
Mathematics Subject Classification ID: 11F06
Mathematics Subject Classification ID: 20H05
Mathematics Subject Classification ID: 20H10
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20J06
Mathematics Subject Classification ID: 19B37
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 5306886
zbMATH Keywords: profinite completions
zbMATH Keywords: cohomological goodness
zbMATH Keywords: Bianchi groups
zbMATH Keywords: limit groups
zbMATH Keywords: cohomology of groups
zbMATH Keywords: central extensions
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: fully residually free groups
zbMATH Keywords: subgroup separable groups
author: Fritz Grunewald
author: Andrei Jaikin-Zapirain
author: Pavel Zalesskij
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2008-031
OpenAlex ID: W2008796784
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description: scientific article; zbMATH DE number 5306886