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Probabilistic nilpotence in infinite groups

Armando Martino, Matthew Tointon, Motiejus Valiunas, Enric Ventura

TL;DR

This work extends the notion of degree of k-step nilpotence to infinite groups by sampling with random walks, Følner sequences, and index-detecting measures. It proves that for finitely generated G, a positive degree implies G is virtually k-step nilpotent, and that the degree is largely independent of the sampling method via polynomial mappings. It also treats residually finite groups, showing that a uniform lower bound on finite-quotient degrees forces a finite-index k-step nilpotent subgroup, and it generalizes Gallagher-type submultiplicativity to finite quotients. The paper combines methods from finite-group nilpotence, polynomial mappings, and profinite/amenable techniques to connect probabilistic identities with classical nilpotence structure.

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.

Probabilistic nilpotence in infinite groups

TL;DR

This work extends the notion of degree of k-step nilpotence to infinite groups by sampling with random walks, Følner sequences, and index-detecting measures. It proves that for finitely generated G, a positive degree implies G is virtually k-step nilpotent, and that the degree is largely independent of the sampling method via polynomial mappings. It also treats residually finite groups, showing that a uniform lower bound on finite-quotient degrees forces a finite-index k-step nilpotent subgroup, and it generalizes Gallagher-type submultiplicativity to finite quotients. The paper combines methods from finite-group nilpotence, polynomial mappings, and profinite/amenable techniques to connect probabilistic identities with classical nilpotence structure.

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.

Paper Structure

This paper contains 17 sections, 58 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. The algebraic structure of probabilistically nilpotent groups
  3. Products of measures that measure index uniformly
  4. Equations over virtually nilpotent groups in terms of polynomial mappings
  5. Sparsity of roots of polynomial mappings
  6. Finite quotients
  7. Submultiplicativity of degree of nilpotence
  8. Sketch of the proof of Proposition \ref{['prop:gallagher']}
  9. Combinatorial structure of Nk(G)
  10. Completing the proof of Proposition \ref{['prop:gallagher']}
  11. Residually finite groups
  12. Dependence on rank
  13. The group Gk(n,r,s)
  14. Non-existence of large k-step nilpotent subgroups
  15. Qualitative conclusions
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $G$ be a finite group such that $\textup{dc}(G)\ge\alpha>0$. Then $G$ has a normal subgroup $\Gamma$ of index at most $\alpha^{-1}+1$ and a normal subgroup $H$ of cardinality at most $\exp(O(\alpha^{-O(1)}))$ such that $H\subset\Gamma$ and $\Gamma/H$ is abelian.

Theorems & Definitions (107)

  • Theorem 1.1: P. M. Neumann neumann
  • Theorem 1.2: Shalev; see the proof of shalev
  • Corollary 1.3: Shalev shalev
  • Definition
  • Theorem 1.4: comm.prob
  • Definition : uniform detection of index
  • Definition : uniform measurement of index
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7
  • ...and 97 more