Stable group theory and approximate subgroups

By Ehud Hrushovski. 2009

225 citations recorded by the source corpus.

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.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/a69d6e697f27bed29304c16f30c5f0437e38214a

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Citations in the corpus

Approximate Subgroups of Linear Groups (2010, methodology): “The idea that the techniques of Larsen and Pink might be useful to us came to us through an analogous application of these ideas by Hrushovski [35] (see also [36]).”
On the diameter of permutation groups (2011, unknown): “The analogy between subgroups and slowly growing sets was also explored in a model-theoretic setting in later work by Hrushovski [Hru12].)”
A Probabilistic Technique for Finding Almost-Periods of Convolutions (2010, methodology): “5 itself was recently used by Green, Sanders and Tao [22] to provide combinatorial proofs of some Freiman-type results of Hrushovski [24].”
Expander graphs, gonality, and variation of Galois representations (2010, background): “Hrushovski [36] are also applicable to obtain gonality growth.”
Growth in finite simple groups of Lie type of bounded rank (2010, methodology): “ of the simple factors appearing in Γ/His bounded by n, and the number of factors is also at most n. This theorem may be viewed as a common generalisation of Theorem 4above and a result of Hrushovski [32]obtained by model-theoretic tools. It would be most interesting to obtain a result that would al”
Growth in groups: ideas and perspectives (2013, unknown): “p theory. Model theory. If Gis inﬁnite, it makes sense to ask how |Ak| grows as k→ ∞. One of the main results here is Gromov’s theorem [Gro81]. The work of Hrushovski et al., which would culminate in [Hru12] (see also [HP95] and [HW08]), used model theory to study subgroups of algebraic groups, reco”
The structure of approximate groups (2011, methodology): “11 we obtain the following result, originally due to Hrushovski [32]).”