The rank of a direct power of a small-cancellation group
type: Item
instance of: scholarly article
title: The rank of a direct power of a small-cancellation group
zbMATH Open document ID: 1022.20013
DOI: 10.1023/A:1020968914280
author: Daniel T. Wise
published in: Geometriae Dedicata
publication date: 2003-02-04T00:00:00Z
review text: As usual, for any group \(G\) denote by \(\text{rank}(G)\) the minimal number of generators of \(G\). Denote by \(G^n\) the \(n\)-th direct power \(G\times\cdots\times G\) of the group \(G\). There are examples of groups \(G\) for which \(\lim_{n\to\infty}\text{rank}(G^n)=\infty\) (take, for example, any finite group \(G\) or any group \(G\) with a proper finite factor), as well as examples for which the numbers \(\text{rank}(G^n)\) have a common finite upper bound (take, for example, any group \(G\) with the condition \(G\cong G\times G\)). The main aim of the current paper is to construct groups \(G\) with small cancellation condition \(C'(1/6)\), such that \(\text{rank}(G^n)\) is bounded. In particular, it is constructed a group \(G_\infty\) which satisfies \(C'(1/6)\), and for which \(\text{rank}((G_\infty)^n)=2\) for all \(n\geq 1\). This group \(G_\infty\) is infinitely presented. But for any \(n\geq 1\) it is possible to find a finitely presented group \(G_n\) which satisfies \(C'(1/6)\), and for which \(\text{rank}((G_n)^n)=2\). Another approach used in the paper is the construction of an epimorphism \(D\to D\times D\) for a group \(D\) with condition \(C'(1/6)\). This automatically provides us with an epimorphism \(D \to D^n\) and, thus, \(\text{rank}(D^n)=\text{rank}(D)\).
Mathematics Subject Classification ID: 20F06
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F67
Mathematics Subject Classification ID: 20E26
Mathematics Subject Classification ID: 20E22
zbMATH DE Number: 1864100
zbMATH Keywords: direct sums
zbMATH Keywords: growth sequences
zbMATH Keywords: groups of bounded rank
zbMATH Keywords: residually finite groups
zbMATH Keywords: small cancellation theory
zbMATH Keywords: word-hyperbolic groups
zbMATH Keywords: numbers of generators
zbMATH Keywords: finitely presented groups
reviewed by: Vahagn H. Mikaelian
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1864100