Growth of permutational extensions

Details Growth of permutational extensions Growth of permutational extensions

2010-11-24

arxiv.org/abs/1011.5266v2

Mathematics

Group Theory

Revision with a few typos fixed. to appear in Inventiones Mathematicae

Scientific paper

10.1007/s00222-011-0368-x

We study the geometry of a class of group extensions, containing permutational wreath products, which we call "permutational extensions". We construct for all natural number k a torsion group with growth function asymptotically $\exp(n^{1-(1-\alpha)^k}),\quad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2$, and a torsion-free group with growth function asymptotically $\exp(\log(n)n^{1-(1-\alpha)^k})$. These are the first examples of groups of intermediate growth for which the growth function is known. We construct a group of intermediate growth that contains the group of finitely supported permutations of a countable set as a subgroup. This gives the first example of a group of intermediate growth containing an infinite simple group as a subgroup.

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