Stable group theory and approximate subgroups

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 G , we show that a finite subset X X with | X X − 1 X | / | X | |X X ^{-1}X |/ |X| bounded is close to a finite subgroup, or else to a subset of a proper algebraic subgroup of G 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.