Abstract. We develop the Plünnecke-Ruzsa and Balog-Szemerédi-Gowers theory of sum set estimates in the non-commutative setting, with discrete, continuous, and metric entropy formulations of these estimates. We also develop a Freiman-type inverse theorem for a special class of 2-step nilpotent groups, namely the Heisenberg groups with no 2-torsion in their vertical group. 1.

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 |XX −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. Modeltheoretically we prove the independence theorem and the stabilizer theorem in a general first-order setting. 1.