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Probabilistic nilpotence in infinite groups

Probabilistic nilpotence in infinite groups
Probabilistic nilpotence in infinite groups
The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Folner sequence if G is amenable. In our first main result we show that if G is finitely generated then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent. This generalises both an earlier result of the second author treating the case k=1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below then G is virtually k-step nilpotent, answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients, generalising a result of Gallagher.
math.GR, math.CO, math.PR
0021-2172
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Tointon, Matthew
721f019c-531b-4b88-99e1-4de8b1c57fce
Valiunas, Motiejus
23b32cdf-14c1-409c-9fc7-d53b54623233
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Tointon, Matthew
721f019c-531b-4b88-99e1-4de8b1c57fce
Valiunas, Motiejus
23b32cdf-14c1-409c-9fc7-d53b54623233
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647

Martino, Armando, Tointon, Matthew, Valiunas, Motiejus and Ventura, Enric (2020) Probabilistic nilpotence in infinite groups. Israel Journal of Mathematics.

Record type: Article

Abstract

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x_1,...,x_{k+1}) in G^{k+1} for which the simple commutator [x_1,...,x_{k+1}] is equal to the identity. In this paper we study versions of this for an infinite group G, with the degree of nilpotence defined by sampling G in various natural ways, such as with a random walk, or with a Folner sequence if G is amenable. In our first main result we show that if G is finitely generated then the degree of k-step nilpotence is positive if and only if G is virtually k-step nilpotent. This generalises both an earlier result of the second author treating the case k=1 and a result of Shalev for finite groups, and uses techniques from both of these earlier results. We also show, using the notion of polynomial mappings of groups developed by Leibman and others, that to a large extent the degree of nilpotence does not depend on the method of sampling. As part of our argument we generalise a result of Leibman by showing that if f is a polynomial mapping into a torsion-free nilpotent group then the set of roots of f is sparse in a certain sense. In our second main result we consider the case where G is residually finite but not necessarily finitely generated. Here we show that if the degree of k-step nilpotence of the finite quotients of G is uniformly bounded from below then G is virtually k-step nilpotent, answering a question of Shalev. As part of our proof we show that degree of nilpotence of finite groups is sub-multiplicative with respect to quotients, generalising a result of Gallagher.

Text
1805.11520v4 - Accepted Manuscript
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More information

Accepted/In Press date: 18 May 2020
e-pub ahead of print date: 18 May 2020
Additional Information: 31 pages. Some changes to theorem numbering
Keywords: math.GR, math.CO, math.PR

Identifiers

Local EPrints ID: 440956
URI: http://eprints.soton.ac.uk/id/eprint/440956
ISSN: 0021-2172
PURE UUID: 1f04d984-314b-4927-b472-8ccf2d93825b
ORCID for Armando Martino: ORCID iD orcid.org/0000-0002-5350-3029
ORCID for Motiejus Valiunas: ORCID iD orcid.org/0000-0003-1519-6643

Catalogue record

Date deposited: 26 May 2020 16:30
Last modified: 18 May 2021 04:01

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Contributors

Author: Armando Martino ORCID iD
Author: Matthew Tointon
Author: Motiejus Valiunas ORCID iD
Author: Enric Ventura

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