Finite generation of iterated wreath products

By Ievgen V. Bondarenko. 2010

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Let (Gn, Xn) be a sequence of finite transitive permutation groups with uniformly bounded number of generators. We prove that the infinitely iterated permutational wreath product $${\ldots\wr G_2\wr G_1}$$ is topologically finitely generated if and only if the profinite abelian group $${\prod_{n\geq 1} G_n/G'_n}$$ is topologically finitely generated. As a corollary, for a finite transitive group G the minimal number of generators of the wreath power $${G\wr \ldots\wr G\wr G}$$ (n times) is bounded if G is perfect, and grows linearly if G is non-perfect. As a by-product we construct a finitely generated branch group, which has maximal subgroups of infinite index.

Subfields: Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/41c758a3ed1985a4f94c3eda1cf769c7765a46ab

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In Computational Theory and Mathematics

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Citations in the corpus

Some topics in the dynamics of group actions on rooted trees (2011, background): “Bondarenko [32] found necessary and sufficient conditions for the infinite iterated permutational wreath product of copies of a finite transitive group of permutations, considered as a profinite group, to be topologically finitely generated.”