Growth of permutational extensions

Abstract

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 $\exp (n^{1-(1-\alpha {)}^k}),2^{3-3/\alpha }+2^{2-2/\alpha }+2^{1-1/\alpha }=2$, 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.

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

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