  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. Combining
these methods with Gromov's proof, we show that a finitely generated group with
an approximate subgroup containing any given finite set must be
nilpotent-by-finite. Model-theoretically we prove the independence theorem and
the stabilizer theorem in a general first-order setting.
