Cockcroft presentations
type: Item
instance of: scholarly article
title: Cockcroft presentations
zbMATH Open document ID: 0851.20029
DOI: 10.1016/0022-4049(95)00027-5
author: C. W. Kilgour
author: Stephen J. Pride
published in: Journal of Pure and Applied Algebra
publication date: 1996-10-14T00:00:00Z
review text: Let \(p\) be a prime or 0. A presentation \(\mathcal P\) is said to be \(p\)-Cockcroft if the map \(\pi_2({\mathcal P})\to H_2({\mathcal P}, Z_p)\) is 0. This concept is related to the minimality and efficiency of presentations and to the relation gap problem. In fact, a finite presentation is efficient iff it is \(p\)-Cockcroft for some prime \(p\). Thus the \(p\)-Cockcroft notion is a natural extension of the efficiency notion to arbitrary (not necessarily finite) presentations. If \(\mathcal P\) is \(p\)-Cockcroft then there exist minimal subgroups in the group being presented for which the corresponding covering complex is \(p\)-Cockcroft (called \(p\)-Cockcroft thresholds). The authors investigate the \(p\)-Cockcroft property and thresholds in terms of combinatorial homotopy (pictures). In particular they show that \(p\)-Cockcroft thresholds are non-unique. Finally, they obtain necessary and sufficient conditions for the presentations of various group constructions (e.g. generalized graph of groups) to be \(p\)-Cockcroft.
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 57M20
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 858647
zbMATH Keywords: \(p\)-Cockcroft presentations
zbMATH Keywords: minimality
zbMATH Keywords: efficiency of presentations
zbMATH Keywords: relation gap problem
zbMATH Keywords: finite presentations
zbMATH Keywords: minimal subgroups
zbMATH Keywords: covering complexes
zbMATH Keywords: \(p\)-Cockcroft thresholds
zbMATH Keywords: combinatorial homotopy
zbMATH Keywords: generalized graphs of groups
MaRDI profile type: Publication
full work available at URL: 0022-4049(95)00027-5
OpenAlex ID: W4210617925
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description: scientific article; zbMATH DE number 858647