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Cohomological goodness and the profinite completion of Bianchi groups. (English) Zbl 1194.20029

In this accessible and carefully written paper, the authors explore the concept of cohomological ‘goodness’, which was introduced by 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 well as limit groups virtually admit a hierarchy, that is, any group of this kind has a finite index subgroup which can be decomposed as a tower of amalgamated free products or HNN-extensions of finitely generated subgroups starting from the trivial group. Moreover the hierarchy of a Bianchi or a limit group behaves well with respect to the profinite topology: since the group is subgroup separable (also known as locally extended residually finite), the hierarchy is preserved in the profinite completion, which allows one to use the Mayer-Vietoris sequence inductively.
Subgroup separability for Bianchi groups and limit groups was established by D. Long and A. W. Reid [Surface subgroups and subgroup separability in \(3\)-manifold topology. Paper from the \(25\)th Brazilian mathematics colloquium – colóquio Brasileiro de matemática, Rio de Janeiro 2005. Rio de Janeiro: IMPA (2005; Zbl 1074.57010)] and by H. Wilton [Geom. Funct. Anal. 18, No. 1, 271-303 (2008; Zbl 1158.20020)], respectively.
In the last two sections of their paper, the authors provide examples of arithmetic subgroups of semisimple groups which are not good and they give a short application of cohomological goodness. Indeed, they prove that certain natural central extensions of Fuchsian groups are residually finite, in contrast to examples of P. Deligne [C. R. Acad. Sci., Paris, Sér. A 287, 203-208 (1978; Zbl 0416.20042)], who showed that analogous central extensions of \(\text{Sp}(4,\mathbb{Z})\) are not residually finite.

MSC:

20E18 Limits, profinite groups
11F75 Cohomology of arithmetic groups
11F06 Structure of modular groups and generalizations; arithmetic groups
20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)
20H10 Fuchsian groups and their generalizations (group-theoretic aspects)
20J05 Homological methods in group theory
20J06 Cohomology of groups
19B37 Congruence subgroup problems
20E26 Residual properties and generalizations; residually finite groups
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