Stable group theory and approximate subgroups

Hrushovski, Ehud¹

¹ Institute of Mathematics, Hebrew University at Jerusalem, Giv’at Ram, 91904 Jerusalem, Israel

Journal of the American Mathematical Society, Tome 25 (2012) no. 1, pp. 189-243

Résumé

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.

Détail

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

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