Amenability and growth of one-relator groups
In the first part of the paper under review, the authors present an algorithm that decides whether a group given by some one-relator presentation is amenable (which is equivalent to soluble in this situation) or not. Given a group \(G\) with finite generating set \(A\), let \(\gamma_A^G(n)\) denote the number of elements of \(G\) that can be written as a product of at most \(n\) of the generators (and their inverses). Then \[ \lambda_A(G)=\lim_{n\to\infty}(\gamma_A^G(n))^{1/n} \] is called the growth rate of \(G\) (with respect to \(A\)). If \(\lambda_A(G)>1\) for some (and then for all) \(A\), \(G\) is said to have exponential growth. Now, let \[ \lambda_*(G)=\liminf_A\lambda_A(G). \] If \(\lambda_*(G)>1\), then \(G\) has uniform exponential growth. The authors conjecture that every one-relator group of exponential growth has in fact uniform exponential growth. In the second part of their paper, they proceed to prove this assertion in a number of cases. This is done by relating the growth rate of \(G\) to the growth of the corresponding graded Lie algebra.
scientific article; zbMATH DE number 1140133