LOG groups and cyclically presented groups
type: Item
instance of: scholarly article
title: LOG groups and cyclically presented groups
zbMATH Open document ID: 0851.20025
author: N. D. Gilbert
author: James Howie
published in: Journal of Algebra
publication date: 1996-11-24T00:00:00Z
review text: A labelled oriented graph (LOG) is a directed graph together with a labelling of its edges by vertices. It determines a group presentation with generators corresponding to the vertices, and relators determined by the edges where the relator says that the edge label conjugates the (generator corresponding to the) initial vertex of the edge to the terminal vertex. The Wirtinger presentation of a classical knot group (with the redundant relator removed) is an LOG presentation for which the underlying graph is a chain. More generally the fundamental group of the complement in \(S^{n+2}\) of any smoothly imbedded, closed, orientable \(n\)-manifold is an LOG group. In this paper, the authors study a class of LOG presentations for which the underlying graph consists of a circuit together with one other edge which has one endpoint a vertex in the circuit. These are endowed with a certain kind of labelling, and the resulting groups are denoted \(G(n,t)\) where \(n\) is the number of vertices in the circuit and \(t\) is determined by the particular labelling. Each \(G(n,t)\) is a semidirect product of its commutator subgroup \(H(n,t)\) with an infinite cyclic group. The \(H(n,t)\) have cyclic presentations analogous to those of the Fibonacci groups. For all but five values of the pair of parameters \((n,t)\), it is determined whether \(H(n,t)\) is finite, and for the finite cases the group is determined. The authors also study the asphericity of the 2-complex \(L(n,t)\) determined by the cyclic presentation of \(H(n,t)\). For most values of \((n,t)\) (including all values with \(n\geq 10\)) except for the families \(L(n,0)\), \(L(n,2)\), \(L(n,n-1)\), \(L(2k,k+1)\), and \(L(2t-1,t)\), \(L(n,t)\) is aspherical, and hence \(H(n,t)\) and \(G(n,t)\) are torsionfree.
reviewed by: Darryl McCullough
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 05C20
zbMATH DE Number: 783856
zbMATH Keywords: labelled oriented graphs
zbMATH Keywords: directed graphs
zbMATH Keywords: group presentations
zbMATH Keywords: generators
zbMATH Keywords: relators
zbMATH Keywords: Wirtinger presentations
zbMATH Keywords: knot groups
zbMATH Keywords: fundamental groups
zbMATH Keywords: cyclic presentations
zbMATH Keywords: Fibonacci groups
zbMATH Keywords: asphericity
zbMATH Keywords: 2-complexes
MaRDI profile type: Publication
full work available at URL: jabr.1995.1119
OpenAlex ID: W2081638221
DOI: 10.1006/JABR.1995.1119
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description: scientific article; zbMATH DE number 783856