SPP 2026 - Subproject: Profinite and RFRS groups

Basic data for this project

Description

What can you say about an infinite group if you know what its finite quotients are? Much recent work has focussed on this question in the context of 3-manifold groups, where the emerging answer is 'quite a lot', but significant gaps in our understanding remain.The goal of this project is to initiate the study of profinite aspects of residually finite rationally solvable (RFRS) groups. RFRS groups were introduced by Ian Agol in his celebrated work on fibering of 3-manifolds.Our primary objective is to show that RFRS groups with the necessary homological finiteness property $FP_\infty$ are good in the sense of Serre, that is, that their cohomology with finite coefficients is naturally isomorphic to that of their profinite completions. In addition to generalizing goodness for virtually compact special groups, and thereby many 3-manifold groups, this has potential applications to profinite rigidity.