Commutator maps, measure preservation, and T-systems

Shelly Garion*, Aner Shalev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

45 Scopus citations

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 ζG (s) = ΣX∈Irr(G) X(1)-s plays a key role in the proofs.

Original languageAmerican English
Pages (from-to)4631-4651
Number of pages21
JournalTransactions of the American Mathematical Society
Volume361
Issue number9
DOIs
StatePublished - Sep 2009

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