Speaker: Martin Bridson, Oxford University
January 4, 2024 02:30 PM
- 03:30 PM
Abstract:
I will begin with an overview of how the study of profinite rigidity has thrived in recent years due to a rich interplay between group theory, low-dimensional geometry and arithmetic. I’ll then describe recent work that underscores the importance of finiteness properties and the homology of groups in this context, describing results that exemplify extremes of rigid and non-rigid phenomena.
A finitely generated, residually finite group G is said to be profinitely rigid if the only finitely generated, residually finite groups with the same set of finite quotients as G are those that are isomorphic to G. One also wants to know which properties P of groups are profinite invariants, i.e. if G has P and H has the same finite quotients as G, does H have P? Sometimes profinite rigidity fails because G admits a Grothendieck pair, i.e. there is a finitely generated subgroup H < G such that the inclusion map induces an isomorphism from the profinite completion of H to the profinite completion of G. The main focus of this talk will be on the art of constructing Grothendieck pairs.