Abstract

The profinite rigidity of a discrete group Γ is intimately related to the profinite rigidity of Γ-module representations. The quest for understanding the profinite rigidity of groups thus calls for a systematic theory of profinite rigidity in algebraic categories. In this talk, I will present a framework for the study of profinite rigidity in the category of modules over a Noetherian domain Λ. I will explore profinite invariants for Λ-modules over any Noetherian domain Λ, as well as absolute profinite rigidity results in the presence of certain homological assumptions on Λ. I will then go on to discuss applications to the profinite rigidity of solvable groups. In particular, I will sketch a proof that solvable Baumslag–Solitar groups and free metabelian groups are profinitely rigid in the absolute sense. These are the first known examples of absolute profinite rigidity among non-abelian one-relator groups and among non-LERF groups.