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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Commutator maps, measure preservation, and $ T$-systems


Authors: Shelly Garion and Aner Shalev
Journal: Trans. Amer. Math. Soc. 361 (2009), 4631-4651
MSC (2000): Primary 20D06, 20P05, 20D60
Published electronically: April 6, 2009
MathSciNet review: 2506422
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Abstract: Let $ G$ be a finite simple group. We show that the commutator map $ \alpha:G \times G \rightarrow G$ is almost equidistributed as $ \vert G\vert \rightarrow\infty$. This somewhat surprising result has many applications. It shows that a for a subset $ X \subseteq G$ we have $ \alpha^{-1}(X)/\vert G\vert^2 = \vert X\vert/\vert G\vert + o(1)$, namely $ \alpha$ 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 $ \vert G\vert \rightarrow \infty$. 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 $ \zeta^G(s) = \sum_{\chi \in \operatorname{Irr}(G)}\chi(1)^{-s}$ plays a key role in the proofs.


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Additional Information

Shelly Garion
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

Aner Shalev
Affiliation: Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04575-9
PII: S 0002-9947(09)04575-9
Received by editor(s): February 21, 2007
Received by editor(s) in revised form: June 24, 2007
Published electronically: April 6, 2009
Additional Notes: The second author acknowledges the support of grants from the Israel Science Foundation and the Bi-National Science Foundation United-States Israel
Article copyright: © Copyright 2009 American Mathematical Society