Growth of permutational extensions

By Laurent Bartholdi, A. Erschler. 2010

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AbstractWe study the geometry of a class of group extensions, containing permutational wreath products, which we call “permutational extensions”. We construct for all k∈ℕ a torsion group Kk with growth function $$v_{K_k}(n)\sim\exp(n^{1-(1-\alpha)^k}),\qquad 2^{3-3/\alpha}+2^{2-2/\alpha}+2^{1-1/\alpha}=2,$$ and a torsion-free group Hk with growth function $$v_{H_k}(n)\sim\exp(\log(n)n^{1-(1-\alpha)^k}).$$ These are the first examples of groups of intermediate growth for which the asymptotics of their 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.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/eb21a7b57fe761e7e08a059bf70604ccc762860a

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Groups of given intermediate word growth (2011, methodology): “nd, there are combinatorial constructions of suﬃciently many elements in the ball of radius R, see [2,11,24]; probabilistic methods have also been used, see [13,15]. A previous article by the authors [6] explores the growth of permutational wreath products of the form A≀X G:“ AX ⋊ G, for a group Aan”