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} ∈ 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 Følner 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 (Theorem 1.5). 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 (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). 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 (Theorem 1.19), 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 (Theorem 1.21), generalising a result of Gallagher.

539-588

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

September 2021

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
(2021)
Probabilistic nilpotence in infinite groups.
*Israel Journal of Mathematics*, 244 (2), .
(doi:10.1007/s11856-021-2168-3).

## Abstract

The ‘degree of k-step nilpotence’ of a finite group G is the proportion of the tuples (x
_{1},…, x
_{k+1} ∈ 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 Følner 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 (Theorem 1.5). 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 (Theorem 1.12). As part of our argument we generalise a result of Leibman by showing that if ϕ is a polynomial mapping into a torsion-free nilpotent group, then the set of roots of ϕ is sparse in a certain sense (Theorem 5.1). 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 (Theorem 1.19), 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 (Theorem 1.21), generalising a result of Gallagher.

Text

** 1805.11520v4
- Accepted Manuscript**
## More information

Accepted/In Press date: 18 May 2020

e-pub ahead of print date: 18 May 2020

Published date: September 2021

Additional Information:
Funding Information:
The fourth author acknowledges partial support from the Spanish Agencia Estatal de Investigación, through grant MTM2017-82740-P (AEI/FEDER, UE), and also from the Graduate School of Mathematics through the “María de Maeztu” Programme for Units of Excellence in R&D (MDM-2014-0445).
Funding Information:
The second author was supported by grant FN 200021_163417/1 of the Swiss National Fund for scientific research.
Publisher Copyright:
© 2021, The Hebrew University of Jerusalem.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

## Identifiers

Local EPrints ID: 440956

URI: http://eprints.soton.ac.uk/id/eprint/440956

ISSN: 0021-2172

PURE UUID: 1f04d984-314b-4927-b472-8ccf2d93825b

## Catalogue record

Date deposited: 26 May 2020 16:30

Last modified: 17 Mar 2024 05:34

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## Contributors

Author:
Matthew Tointon

Author:
Motiejus Valiunas
Author:
Enric Ventura

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