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Mathematics > Logic
Title: Stable group theory and approximate subgroups
(Submitted on 11 Sep 2009 (v1), last revised 16 May 2011 (this version, v4))
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 |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.
Submission history
From: Ehud Hrushovski [view email][v1] Fri, 11 Sep 2009 18:25:44 GMT (49kb)
[v2] Fri, 18 Sep 2009 04:07:29 GMT (50kb)
[v3] Tue, 24 Aug 2010 15:14:28 GMT (70kb)
[v4] Mon, 16 May 2011 13:12:31 GMT (75kb)
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