×

Further groups that do not have uniformly exponential growth. (English) Zbl 1133.20034

From the introduction: Let \(G\) be a finitely generated group and let \(S\) be a finite generating set. For each integer \(n\geq 0\), let \(\gamma_S(n)\) be the number of elements of \(G\) that are products of at most \(n\) elements of \(S\cup S^{-1}\), and let \(e_S(G)=\lim_{n\to\infty}(\gamma_S(n))^{1/n}\). This number is called the ‘exponential growth rate’ of \(G\) with respect to \(S\). If \(e_S(G)>1\) for one generating set \(S\) then \(e_S(G)>1\) for all generating sets \(S\), and then \(G\) is said to have ‘exponential growth’.
In [Invent. Math. 155, No. 2, 287-303 (2004; Zbl 1065.20054)] we constructed groups \(G\) having non-Abelian free subgroups and having a sequence \((S_n)\) of generating sets such that \(e_{S_n}\to 1\) as \(n\to \infty\), answering a question of Gromov. Here we generalize the examples in [loc. cit.] and propose a different approach to the existence of non-Abelian free subgroups.
Theorem 1. Let \(d\) be a positive integer and let \(\mathcal X\) be a class of groups with the following properties: (i) each group in \(\mathcal X\) is perfect and can be generated by \(d\) involutions; (ii) each group \(G\in\mathcal X\) is isomorphic to a permutational wreath product \(H\text{\,wr\,}A_m\) where \(H\in\mathcal X\) and \(m\geq 29\). Then each group \(G\in \mathcal X\) has two sequences \((x_n)\), \((y_n)\) of elements such that (a) \(x_n^2=y_n^3=1\) and \(\langle x_n,y_n\rangle=G\) for each \(n\), and (b) \(e_{\{x_n,y_n\}}(G)\to 1\) as \(n\to\infty\).

MSC:

20F69 Asymptotic properties of groups
20F05 Generators, relations, and presentations of groups
20E22 Extensions, wreath products, and other compositions of groups
20F65 Geometric group theory
20E07 Subgroup theorems; subgroup growth

Citations:

Zbl 1065.20054
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Grigorchuk, R. I., Just infinite branch groups, (New Horizons in Pro-\(p\) Groups (2000), Birkhäuser), 121-179 · Zbl 0982.20024
[2] Gromov, M., Structures métriques pour les variétés riemanniennes (1981), CEDIC: CEDIC Paris · Zbl 0509.53034
[3] Milnor, J., A note on curvature and the fundamental group, J. Differential Geom., 2, 1-7 (1968) · Zbl 0162.25401
[4] Neumann, P. M., Some questions of Edjvet and Pride about infinite groups, Illinois J. Math., 30, 301-316 (1986) · Zbl 0598.20029
[5] Segal, D., The finite images of finitely generated groups, Proc. London Math. Soc. (3), 82, 597-613 (2001) · Zbl 1022.20011
[6] Tamburini, C.; Wilson, J. S., A residual property of certain free products, Math. Z., 186, 525-530 (1984) · Zbl 0545.20019
[7] Tamburini, M. C.; Wilson, J. S., On the (2,3)-generation of automorphism groups of free groups, Bull. London Math. Soc., 29, 43-48 (1997) · Zbl 0874.20018
[8] Wilson, J. S., Economical generating sets for finite simple groups, (Groups of Lie Type and Their Geometries (1995), Cambridge University Press), 289-302 · Zbl 0846.20034
[9] Wilson, J. S., On exponential growth and uniformly exponential growth for groups, Invent. Math., 155, 287-303 (2004) · Zbl 1065.20054
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.