Commutator maps, measure preservation, and 𝑇-systems

Abstract

Abstract. Let G be a finite simple group. We show that the commutator map α : G × G → G is almost equidistributed as |G| → ∞. This somewhat surprising result has many applications. It shows that a for a subset X ⊆ G we have α −1 (X)/|G| 2 = |X|/|G| + o(1), namely α is almost measure preserving. From this we deduce that almost all elements g ∈ G can be expressed as commutators g = [x, y] where x, y generate G.This enables us to solve some open problems regarding T -systems and the Product Replacement Algorithm (PRA) graph. We show that the number of T -systems in G with two generators tends to infinity as |G| → ∞. This settles a conjecture of Guralnick and Pak. A similar result follows for the number of connected components of the PRA graph of G with two generators.Some of our results apply for more general finite groups and more general word maps.Our methods are based on representation theory, combining classical character theory with recent results on character degrees and values in finite simple groups. In particular the so called Witten zeta function ζ G (s) = χ∈Irr(G) χ(1) −s plays a key role in the proofs.

The Waring problem for finite simple groups

“…This result is new even for k = 2; in fact, it is likely that the word x 2 1 x 2 2 is surjective on all finite simple groups of order > 2 (see [GS09] for related results). Finally, our methods may be relevant to a well-known conjecture of J. G. Thompson, stating that any finite simple group Γ has a conjugacy class C such that C 2 = Γ.…”

The Ore conjecture

“…This implies that every element of a large finite simple group is a product of two commutators. In [19] it is shown that the commutator map on finite simple groups is almost measure-preserving, a result having applications to the product replacement algorithm [7].…”

Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics

“…Note that we cannot require in this theorem that S = G. Indeed, it is well known (and follows from (19) Two proofs are given in [41] for this theorem. The first is probabilistic whereas in the second the subsets S are explicitly constructed.…”

“…Yet another type of problems, first introduced in [41], arises when one asks about the behaviour of the fibres of the word map rather than about its image. Namely, Shalev asked [113, Problem 2.10] whether (the cardinalities of) these fibres are equidistributed (or close to be equidistributed) when varies in some "large" subset of the image of and G runs over some family of finite groups.…”

“…Namely, Shalev asked [113, Problem 2.10] whether (the cardinalities of) these fibres are equidistributed (or close to be equidistributed) when varies in some "large" subset of the image of and G runs over some family of finite groups. It was proved in [41] that the word = 2 2 ∈ F 2 , the commutator word = [ ] ∈ F 2 , as well as the words = [ 1 ] ∈ F , -fold commutators in any arrangement of brackets, are almost equidistributed on the family of finite simple non-abelian groups. The case G = SL(2 ) was studied in some more detail in [8,9,14], see Section 6.…”