Pro-\(p\) completions of Poincaré duality groups.
type: Item
instance of: scholarly article
title: Pro-\(p\) completions of Poincaré duality groups.
zbMATH Open document ID: 1312.20025
author: Dessislava H. Kochloukova
author: Igor Lima
author: Jonathan A. Hillman
published in: Israel Journal of Mathematics
publication date: 2014-06-25T00:00:00Z
review text: The goal of the paper under review is to study conditions that ensure that the pro-\(p\) completion of a \(\text{PD}_n\)-group is virtually a pro-\(p\) \(\text{PD}_m\)-group, for some \(m<n-1\). Here \(p\) is a prime, and a group is said to be \(\text{PD}_n\) if it is an orientable Poincaré duality group of dimension \(n\). In particular, it is shown that if (i) \(G\) is an abstract orientable \(\text{PD}_n\) group, for some \(n\geq 3\), (ii) the pro-\(p\) completion \(\widehat G_p\) of \(G\) is infinite and not a virtual orientable pro-\(p\) \(\text{PD}_n\) group, and (iii) for \(2\leq i<n\) the inverse limit (over a directed set of subgroups \(U\) of \(p\)-power index in \(G\) which induces the pro-\(p\) topology of \(G\)) of the \(i\)-th homology group of \(U\) with coefficients in the field \(F\) with \(p\) elements has finite dimension over \(F\), then \(\widehat G_p\) is virtually a pro-\(p\) \(\text{PD}_m\)-group for some \(m<n-1\).
Mathematics Subject Classification ID: 20E18
Mathematics Subject Classification ID: 57P10
Mathematics Subject Classification ID: 20J05
Mathematics Subject Classification ID: 20E26
zbMATH DE Number: 6308411
zbMATH Keywords: pro-\(p\) groups
zbMATH Keywords: pro-\(p\)-completions
zbMATH Keywords: Poincaré duality spaces
zbMATH Keywords: Poincaré duality groups
zbMATH Keywords: homology groups
reviewed by: Andrea Caranti
MaRDI profile type: Publication
full work available at URL: s11856-013-0074-z
OpenAlex ID: W2015311623
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DOI: 10.1007/S11856-013-0074-Z
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description: scientific article; zbMATH DE number 6308411