Commutator maps, measure preservation, and T-systems

By Shelly Garion, Aner Shalev. 2007

74 citations recorded by the source corpus.

Let G be a finite simple group. We show that the commutator map : G ◊G ! G is almost equidistributed as |G| ! 1. This somewhat surprising result has many applications. It shows that a for a subset X G we have 1 (X)/|G| 2 = |X|/|G| + o(1), namely is almost measure preserving. From this we deduce that almost all elements g 2 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| ! 1. 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 G (s) = P 2Irr(G) (1) s plays a key role in the proofs.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6c2777dad73479b1355a00ae86aef1bf0c99496a

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Characters of symmetric groups: sharp bounds and applications (2008, background): “Indeed, a function which is almost uniformly distributed in the L 1-norm is almost measure preserving. See for instance [ GSh ] for more details.”
Generators and commutators in finite groups; abstract quotients of compact groups (2011, unknown): “α,β(X) = Yδ1 i=1 Tα i,β i(S,S) = Yδ1 i=1 γ(S×S)ti = γ(S×S)∗δ1 ·t= S, where t= t1 ...tδ1. The result follows. 4.1.3 Inner automorphisms The key to this case is a result is due to Garion and Shalev. In [GaSh] they deﬁne for each ﬁnite group Gthe invariant ǫ(G) = ζG(2) −1 1/4 , where ζG(2) = P χ(1)−2 s”