On exponential growth and uniformly exponential growth for groups.
The paper gives a negative answer to \textit{M. Gromov}'s question [Structures métriques pour les variétés riemanniennes. Textes Mathématiques, 1. Paris (1981; Zbl 0509.53034)], (Remark 5.2) posed in 1981. Gromov asked `if groups of exponential growth necessarily have uniform exponential growth'. Here are some background information and definitions: for the element \(g\) of the group \(G\) generated by the finite set \(S\) the word length \(l_S(g)\) is the smallest integer \(n\) such that \(g\) is a product of \(n\) elements of \(S\cup S^{-1}\). For each \(n\geq 0\) define \(\gamma_S(n)=|\{g \in G\mid l_S(g)\leq n\}|\). The limit \(e_S(G)=\lim_{n\to \infty}(\gamma_S(n))^{1/n}\) exists and is called the `exponential growth rate' of \(G\) with respect to \(S\). If it is greater than \(1\) for one generating set, then it is greater than \(1\) for all generating sets, and \(G\) is said to have `exponential growth'. Moreover, a group has `uniformly exponential growth' if \(\inf_Se_S(G)>1\), where the infimum is taken over all finite generating sets \(S\) in \(G\). It has been shown that Gromov's question has a positive answer for groups of certain types, such as hyperbolic groups, soluble groups, linear groups in characteristic \(0\), etc. (see [\textit{P. de la Harpe}, Geom. Dedicata 95, 1-17 (2002; Zbl 1025.20027)] for a survey). Nevertheless, a negative answer to Gromov's question follows from the following theorem and from the fact that a finitely generated group having non-Abelian free subgroups must have exponential growth. Theorem 1. There is a group \(G\) which has non-Abelian free subgroups and a sequence \((S_n)\) of generating sets such that \(\lim_{n\to\infty}e_{S_n}(G)=1\). Each generating set \(S_n\) has two elements, one of order 2 and the other of order~3. Let \(A_{31}\) be the alternating group of degree \(31\). The wreath product mentioned below is a `permutational' wreath product. A larger class of groups that do not have uniformly exponential growth is provided by: Theorem 2. Let \(H\) be any finitely generated perfect group with the property that \(H\cong H\text{\,Wr\,}A_{31}\). Then \(H\) has a sequence \((x_n)\) of elements of order \(2\), and a sequence \((y_n)\) of elements of order \(3\) such that: (i) \(\langle x_n,y_n\rangle=H\) for each \(n\), and (ii) \(\lim_{n\to\infty}e_{\{x_n,y_n\}}(H)=1\).
scientific article; zbMATH DE number 2101714