Stable group theory and approximate subgroups

Hrushovski, Ehud

doi:10.1090/S0894-0347-2011-00708-X

Stable group theory and approximate subgroups
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by Ehud Hrushovski

J. Amer. Math. Soc. 25 (2012), 189-243

DOI: https://doi.org/10.1090/S0894-0347-2011-00708-X

Published electronically: June 15, 2011

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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 $|X X ^{-1}X |/ |X|$ 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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