Uniform growth in groups of exponential growth (Q1863076)

Let \(\Gamma\) be a group equipped with a finite system of generators \(S\) and with the corresponding word metric. The associated `growth function' is the function, defined on the set of natural numbers, which assigns to each integer \(k\geq 0\) the quantity denoted by \(\beta(\Gamma,S;k)\) which is equal to the number of elements \(\gamma\in\Gamma\) whose length is \(\leq k\). The `exponential growth rate' is then defined as \[ w(\Gamma,S)=\lim_{k\to\infty}\beta(\Gamma,S;k)^{{1\over k}}. \] The `uniform exponential growth rate' of a finitely-generated group \(\Gamma\) is then defined as \[ w(\Gamma)=\inf\{w(\Gamma,S)\mid S\text{ is a finite set of generators of }\Gamma\}. \] The group \(\Gamma\) is said to be of `exponential growth' if \(w(\Gamma,S)>1\) for some (or equivalently for any) finite generating set \(S\), and the group is said to be of `uniform exponential growth' if \(w(\Gamma)>1\). An open question (asked by Gromov) is whether there exists a finitely-generated group of exponential growth which is not of uniform exponential growth. In the paper under review, the author gives a rich variety of examples of finitely-generated groups which have uniform exponential growth, and he formulates several related conjectures and open problems. The examples that he gives include the following: non-Abelian free groups, semi-direct products of free Abelian groups with automorphisms having an eigenvalue of modulus distinct from 1, Golod-Shafarevich infinite finitely-generated \(p\)-groups, groups which virtually have non-Abelian free quotients, nonelementary hyperbolic groups, soluble groups of exponential growth and appropriate free products with amalgamation, HNN-extensions, and one relator groups of exponential growth.