Bieri-Eckmann criteria for profinite groups
Bieri-Eckmann criteria for profinite groups
In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP n 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 FP n is closed under extensions, quotients by subgroups of type FP n , 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 FP n but not of type FP n+1 over Z ? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.
857-893
Corob Cook, Ged
de8c4e37-ab72-4e7c-ad10-a940066433c5
May 2016
Corob Cook, Ged
de8c4e37-ab72-4e7c-ad10-a940066433c5
Abstract
In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP n 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 FP n is closed under extensions, quotients by subgroups of type FP n , 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 FP n but not of type FP n+1 over Z ? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.
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Bieri-Eckmann criteria for profinite groups.pdf
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Accepted/In Press date: 11 March 2015
e-pub ahead of print date: 26 May 2016
Published date: May 2016
Organisations:
Pure Mathematics
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Local EPrints ID: 397234
URI: http://eprints.soton.ac.uk/id/eprint/397234
ISSN: 0021-2172
PURE UUID: 7134fd9e-d99f-4063-b7a6-64716dc7ea26
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Date deposited: 30 Jun 2016 09:25
Last modified: 15 Mar 2024 05:41
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Author:
Ged Corob Cook
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