Duke Mathematical Journal

Cohomological goodness and the profinite completion of Bianchi groups

F. Grunewald, A. Jaikin-Zapirain, and P. A. Zalesskii

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Abstract

The concept of cohomological goodness was introduced by J.-P. Serre in his book on Galois cohomology [31]. This property relates the cohomology groups of a group to those of its profinite completion. We develop properties of goodness and establish goodness for certain important groups. We prove, for example, that the Bianchi groups (i.e., the groups ${\rm PSL}(2,{\cal O})$, where ${\cal O}$ is the ring of integers in an imaginary quadratic number field) are good. As an application of our improved understanding of goodness, we are able to show that certain natural central extensions of Fuchsian groups are residually finite. This result contrasts with examples of P. Deligne [5], who shows that the analogous central extensions of ${\rm Sp}(4,\mathbb{Z})$ do not have this property

Article information

Source
Duke Math. J. Volume 144, Number 1 (2008), 53-72.

Dates
First available in Project Euclid: 2 July 2008

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1215032810

Digital Object Identifier
doi:10.1215/00127094-2008-031

Mathematical Reviews number (MathSciNet)
MR2429321

Zentralblatt MATH identifier
05306886

Subjects
Primary: 20H05: Unimodular groups, congruence subgroups [See also 11F06, 19B37, 22E40, 51F20] 11F75: Cohomology of arithmetic groups
Secondary: 14G32: Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) 19B37: Congruence subgroup problems [See also 20H05] 57N10: Topology of general 3-manifolds [See also 57Mxx]

Citation

Grunewald, F.; Jaikin-Zapirain, A.; Zalesskii, P. A. Cohomological goodness and the profinite completion of Bianchi groups. Duke Math. J. 144 (2008), no. 1, 53--72. doi:10.1215/00127094-2008-031. http://projecteuclid.org/euclid.dmj/1215032810.


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