Pyber, László and Szabó, Endre (2016) Growth in finite simple groups of Lie type. JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, 29. pp. 95146. ISSN 08940347
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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 wellknown 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$.  See more at: http://www.ams.org/journals/jams/000000000/S089403472014008213/home.html#sthash.Lp65MZge.dpuf
Item Type:  Article 

Uncontrolled Keywords:  GROWTH; Finite simple groups; ALGEBRAIC GROUPS 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika > QA72 Algebra / algebra 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  04 Jan 2017 14:22 
Last Modified:  09 Jan 2017 08:39 
URI:  http://real.mtak.hu/id/eprint/44517 
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