Commutator maps, measure preservation, and T-systems

TL;DR

This paper proves that the commutator map in finite simple groups is almost measure-preserving as group size increases, leading to new insights into T-systems, the PRA graph, and solutions to open problems in group theory.

Contribution

It establishes the near measure-preserving property of the commutator map in large finite simple groups and applies this to solve open problems on T-systems and PRA graph connectivity.

Findings

Commutator map is almost equidistributed in large finite simple groups.

Number of T-systems with two generators tends to infinity as group size increases.

Connected components of the PRA graph with two generators also tend to infinity.

Abstract

Let G be a finite simple group. We show that the commutator map $a : G \times G \to G$ is almost equidistributed as the order of G goes to infinity. This somewhat surprising result has many applications. It shows that for a subset X of G we have $a^{-1}(X)/|G|^2 = |X|/|G| + o(1)$, namely $a$ is almost measure preserving. From this we deduce that almost all elements $g \in G$ 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 the order of G goes to infinity. 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 plays a key role in the proofs.