Iterated identities and iterational depth of groups

Anna  Erschler

doi:10.3934/jmd.2015.9.257

Article Contents

2015, Volume 9: 257-284. Doi: 10.3934/jmd.2015.9.257

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Iterated identities and iterational depth of groups

Anna Erschler^1,

1.
Département de Mathématiques et Applications, École Normale Supérieure, 45 Rue d’Ulm, 75005 Paris

Received: August 31, 2014

Revised: June 30, 2015

Published: August 2015

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Abstract

Given a word $w$ on $n$ letters, we study groups which satisfy ``iterated identity'' $w$, meaning that for all $x_1, \dots, x_n$ there exists $N$ such that the $N-th$ iteration of $w$ of Engel type, applied to $x_1, \dots, x_n$, is equal to the identity. We define bounded groups and groups which are multiscale with respect to identities. This notion of being multiscale can be viewed as a self-similarity conditions for the set of identities, satisfied by a group. In contrast with torsion groups and Engel groups, groups which are multiscale with respect to identities appear among finitely generated elementary amenable groups. We prove that any polycyclic, as well as any metabelian group is bounded, and we compute the iterational depth for various wreath products. We study the set of iterated identities satisfied by a given group, which is not necessarily a subgroup of a free group and not necessarily invariant under conjugation, in contrast with usual identities. Finally, we discuss another notion of iterated identities of groups, which we call solvability type iterated identities, and its relation to elementary classes of varieties of groups.

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Mathematics Subject Classification: Primary: 20E10, 20F69 ; Secondary: 20E22, 43A07, 20F16, 20F18, 20F19, 20F45, 20F50.

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