Galois cohomology of completed link groups

TL;DR

This paper computes the Galois cohomology of pro-p completions of primitive link groups, showing it aligns with the cohomology of the discrete group, and applies this to verify conjectures for their extensions.

Contribution

It provides the first computation of Galois cohomology for pro-p completions of primitive link groups and links this to important conjectures in topology and operator algebras.

Findings

Galois cohomology is isomorphic to discrete group cohomology with Z/pZ coefficients.

Results hold for groups with irreducible linking number diagrams modulo p.

Applications confirm Baum-Connes and Atiyah conjectures for extensions.

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 the 3-sphere whose linking number diagram is irreducible modulo p (e.g. none of the linking numbers is divisible by p). The result is that (with Z/pZ-coefficients) the Galois cohomology is naturally isomorphic to the Z/pZ-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.