The profinite completion of a discrete group encapsulates the information of all finite quotients of the group. Residually finite rationally solvable (RFRS) groups were introduced by Ian Agol in his work on fibring of 3-manifolds and include all subgroups of right-angled Artin groups. This project investigates profinite aspects of RFRS groups with the goal showing that, with sufficient homological finiteness assumptions, RFRS groups are good in the sense of Serre, that is, the inclusion into the profinite completion induces isomorphism on cohomology with finite coefficients.
The unit conjecture, commonly attributed to Kaplansky, predicts that if \(K\) is a field and \(G\) is a torsion-free group then the only units of the group ring \(K[G]\) are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.
48Profinite and RFRS groups