2009/09/11 by Hrushovski, Ehud · 8 citations
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.0909.2190
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.