On the topology of the groups of type \(\mathfrak{Z}\)
type: Item
instance of: scholarly article
title: On the topology of the groups of type \(\mathfrak{Z}\)
zbMATH Open document ID: 1486.57021
author: Kirk McDermott
published in: Topology and its Applications
publication date: 2022-01-26T00:00:00Z
review text: A group \(G\) admits a cyclic presentation \(P\) if the relators are mapped to each other by certain automorphisms of the free group of a generating set of \(G\). If this generating set consists of \(n\) generators then there is a semi-direct product \(E=G\rtimes C_n\) called the shift extension of \(G\). A spherical picture over a presentation of \(E\) lifts to one over \(P\). In this paper it is shown how this picture determines a Heegaard diagram for a 3-manifold inducing \(P\). The method is demonstrated with the groups of type \(\mathfrak Z\), an infinite family of cyclically presented groups whose shift extensions split over centrally extended triangle groups. The resulting manifolds are examples of Dunwoody manifolds and break down into two subfamilies.
reviewed by: Stephan Rosebrock
Mathematics Subject Classification ID: 57K30
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 57M07
zbMATH DE Number: 7464211
zbMATH Keywords: \(3\)-manifold spine
zbMATH Keywords: cyclically presented group
zbMATH Keywords: asphericity
zbMATH Keywords: Heegaard diagram
zbMATH Keywords: triangle group
describes a project that uses: tools-for-cyclically-presented-groups
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107767
OpenAlex ID: W3181924227
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DOI: 10.1016/J.TOPOL.2021.107767
description: scientific article; zbMATH DE number 7464211