# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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## Growth in finite simple groups of Lie typeHTML articles powered by AMS MathViewer

by László Pyber and Endre Szabó
J. Amer. Math. Soc. 29 (2016), 95-146
DOI: https://doi.org/10.1090/S0894-0347-2014-00821-3
Published electronically: October 31, 2014

## Abstract:

We prove that if $L$ is a finite simple group of Lie type and $A$ a set of generators of $L$, then either $A$ grows, i.e., $|A^3| > |A|^{1+\varepsilon }$ where $\varepsilon$ depends only on the Lie rank of $L$, or $A^3=L$. This implies that for simple groups of Lie type of bounded rank a well-known conjecture of Babai holds, i.e., the diameter of any Cayley graph is polylogarithmic. We also obtain new families of expanders.

A generalization of our proof yields the following. Let $A$ be a finite subset of $SL(n,\mathbb {F})$, $\mathbb {F}$ an arbitrary field, satisfying $\big |A^3\big |\le \mathcal {K}|A|$. Then $A$ can be covered by $\mathcal {K}^m$, i.e., polynomially many, cosets of a virtually soluble subgroup of $SL(n,\mathbb {F})$ which is normalized by $A$, where $m$ depends on $n$.

References
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Bibliographic Information
• László Pyber
• Affiliation: A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary
• Email: pyber.laszlo@renyi.mta.hu
• Endre Szabó
• Affiliation: A. Rényi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, H-1364 Budapest, Hungary
• Email: szabo.endre@renyi.mta.hu
• Received by editor(s): June 11, 2014
• Received by editor(s) in revised form: September 15, 2014
• Published electronically: October 31, 2014
• Additional Notes: The first author is supported in part by OTKA 78439 and K84233
The second author is supported in part by OTKA NK81203, K84233 and by the MTA Rényi “Lendület” Groups and Graphs Research Group