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Growth in finite simple groups of Lie type


Authors: László Pyber and Endre Szabó
Journal: J. Amer. Math. Soc. 29 (2016), 95-146
MSC (2010): Primary 20F69; Secondary 20G15, 20D06
DOI: https://doi.org/10.1090/S0894-0347-2014-00821-3
Published electronically: October 31, 2014
MathSciNet review: 3402696
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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., $ \vert A^3\vert > \vert A\vert^{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 \vert A^3\big \vert\le \mathcal {K}\vert A\vert$. 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$.


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Additional 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

DOI: https://doi.org/10.1090/S0894-0347-2014-00821-3
Keywords: Growth, finite simple groups, algebraic groups
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
Article copyright: © Copyright 2014 American Mathematical Society