Galois cohomology of completed link groups

Blomer, Inga; Linnell, Peter; Schick, Thomas

doi:10.1090/S0002-9939-08-09395-7

Galois cohomology of completed link groups
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by Inga Blomer, Peter A. Linnell and Thomas Schick PDF

Proc. Amer. Math. Soc. 136 (2008), 3449-3459 Request permission

Abstract:

In this paper we compute the Galois cohomology of the pro-$p$ completion of primitive link groups. Here, a primitive link group is the fundamental group of a tame link in $S^3$ whose linking number diagram is irreducible modulo $p$ (e.g. none of the linking numbers is divisible by $p$).

The result is that (with $\mathbb {Z}/p\mathbb {Z}$-coefficients) the Galois cohomology is naturally isomorphic to the $\mathbb {Z}/p\mathbb {Z}$-cohomology of the discrete link group.

The main application of this result is that for such groups the Baum-Connes conjecture or the Atiyah conjecture are true for every finite extension (or even every elementary amenable extension), if they are true for the group itself.

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