The rank of a direct power of a small-cancellation group
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, \(\text{rank}(D^n)=\text{rank}(D)\).
scientific article; zbMATH DE number 1864100