Cohomological goodness and the profinite completion of Bianchi groups
F. Grunewald, A. Jaikin-Zapirain, and P. A. Zalesskii
Source: Duke Math. J. Volume 144, Number 1
(2008), 53-72.
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
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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
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