Bieri–Eckmann criteria for profinite groups

Abstract

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FPn over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FPn is closed under extensions, quotients by subgroups of type FPn, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FPn but not of type FPn+1 over ZĈ for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FPn fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.