Families of group presentations related to topology
type: Item
instance of: scholarly article
title: Families of group presentations related to topology
zbMATH Open document ID: 1066.57003
author: Alberto Cavicchioli
author: Dušan Repovš
author: Fulvia Spaggiari
published in: Journal of Algebra
publication date: 2005-04-21T00:00:00Z
review text: The authors consider a class of cyclically presented groups \(G_n^\varepsilon (m,k,h)\), where \(\varepsilon =(a,b,r,s)\in\mathbb Z^4\) for \(n\geq 2\) and the integers \(m,k,h\) are taken modulo \(n\). Then \(G_n^\varepsilon (m,k,h)\) has generators \(x_1,\ldots ,x_n\) and defining relations \[ x_i^a x_{i+k}^b x_{i+h+m}^a = (x_{i+h}^r x_{i+m}^r)^s \] for \(i=1,\ldots ,n\) (subscripts mod \(n\)). This class of groups contains many well known classes of groups like some Fibonacci-groups, Sieradski groups, Gilbert-Howie groups and others. Under certain conditions on the parameters some of these groups are shown to be isomorphic. The main result is the following: Theorem 3.1 Suppose that \(n\) and \(b\) are odd and \(n\) is coprime with \(2k-h-m\). Then the group \(G_n^\varepsilon (m,k,h)\) cannot be the fundamental group of any hyperbolic 3-orbifold of finite volume. In the last part of the paper the authors study a certain subclass of the groups \(G_n^\varepsilon (m,k,h)\), where \(a=b=1\) and \(s=0\). Under certain conditions these groups are shown to be aspherical.
reviewed by: Stephan Rosebrock
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 2159736
zbMATH Keywords: presentation
zbMATH Keywords: cyclically presented groups
zbMATH Keywords: Fibonacci groups
zbMATH Keywords: hyperbolic manifolds
zbMATH Keywords: asphericity
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2005.01.008
OpenAlex ID: W2048404407
cites work: Q4434996
cites work: Q3770791
cites work: Aspherical relative presentations
cites work: Q4399371
cites work: Q3840289
cites work: TOPOLOGICAL PROPERTIES OF CYCLICALLY PRESENTED GROUPS
cites work: Cyclic branched coverings of 2-bridge knots
cites work: LOG groups and cyclically presented groups
cites work: Some Finitely Presented Non-3-Manifold Groups
cites work: Q4381884
cites work: A Finitely Generated Infinite Simple Group
cites work: Q4028057
cites work: Fractional Fibonacci groups and manifolds
cites work: Q4867088
cites work: Generalised Fibonacci manifolds
cites work: Einige endliche Gruppen mit drei Erzeugenden und drei Relationen
cites work: An essay on free products of groups with amalgamations
cites work: Combinatorial squashings, 3-manifolds, and the third homology of groups
cites work: On the presentations of the fundamental groups of 3-manifolds
DOI: 10.1016/J.JALGEBRA.2005.01.008
description: scientific article; zbMATH DE number 2159736