Probabilistic nilpotence in infinite groups

Abstract

The 'degree of k-step nilpotence' of a finite group G is the proportion of the tuples (x1, . . . , x k+1 ) ∈ G k+1 for which the simple commutator [x1, . . . , 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 1.20), generalising a result of Gallagher. Contents 12 6. Finite quotients 16 7. Dependence on rank 23 Appendix A. Polynomial mappings into torsion-free nilpotent groups 27 Appendix B. Hyperbolic groups 28 References 30

Compact groups with a set of positive Haar measure satisfying a nilpotent law

“…Then G has a unique normalized Haar measure denoted by m G . The following question is proposed in [4]. Positive answer to Question 1.1 is known for k = 1 (see [3,Theorem 1.2]).…”

“…Positive answer to Question 1.1 is known for k = 1 (see [3,Theorem 1.2]). It follows from [4,Theorem 1.19] that Question 1.1 has positive answer for arbitrary k whenever we further assume that G is totally disconnected i.e. G is a profinite group.…”

“…

The following question is proposed in [4, Question 1.20]. Let G be a compact group, and suppose thatThe case k = 1 is already known.

…”

Probabilistically-like nilpotent groups

“…He proved the following: for any k ≥ 2 and any ǫ > 0, there exists some r = r(k, ǫ) such that given a finite group G with Pr G,w k (1) ≥ ǫ, there is a characteristic nilpotent subgroup N of G of class at most k such that the group G/N has exponent less than r. This result extends to arbitrary residually finite groups, by taking the probability in their profinite completion (cf. [17,Theorem 1.1 & Corollary 1.4] as well as [11,Theorem 1.30]).…”

Compact groups with a set of positive Haar measure satisfying a nilpotent law

“… Abstract The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that has positive Haar measure in . Does G have an open k -step nilpotent subgroup? We give a positive answer for . …”