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Stable group theory and approximate subgroups


Author: Ehud Hrushovski
Journal: J. Amer. Math. Soc. 25 (2012), 189-243
MSC (2010): Primary 11P70, 03C45
DOI: https://doi.org/10.1090/S0894-0347-2011-00708-X
Published electronically: June 15, 2011
MathSciNet review: 2833482
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Abstract: We note a parallel between some ideas of stable model theory and certain topics in finite combinatorics related to the sum-product phenomenon. For a simple linear group $ G$, we show that a finite subset $ X$ with $ \vert X X ^{-1}X \vert/ \vert X\vert$ bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of $ G$. We also find a connection with Lie groups, and use it to obtain some consequences suggestive of topological nilpotence. Model-theoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting.


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Additional Information

Ehud Hrushovski
Affiliation: Institute of Mathematics, Hebrew University at Jerusalem, Giv’at Ram, 91904 Jerusalem, Israel
Email: ehud@math.huji.ac.il

DOI: https://doi.org/10.1090/S0894-0347-2011-00708-X
Received by editor(s): August 24, 2010
Received by editor(s) in revised form: May 16, 2011
Published electronically: June 15, 2011
Additional Notes: Research supported in part by Israel Science Foundation grant 1048/07.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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