Growth in finite simple groups of Lie type of bounded rank

By László Pyber, E. Szab'o. 2010

181 citations recorded by the source corpus.

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^{1+epsilon} where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type of bounded rank the diameter of any Cayley graph is polylogarithmic in |L|. We obtain a similar bound for the diameters of all Cayley graphs of perfect subgroups of GL(n,p) generated by their elements of order p. We also obtain some new families of expanders. We also prove the following partial extension. Let G be a subgroup of GL(n,p), p a prime, and S a symmetric set of generators of G satisfying |S^3|\le K|S| for some K. Then G has two normal subgroups H\ge P such that H/P is soluble, P is contained in S^6 and S is covered by K^c cosets of H where c depends on n. We obtain results of similar flavour for sets generating infinite subgroups of GL(n,F), F an arbitrary field.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6e0e0e93f041ff640a2570db0e591a1e0f2fe631

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Citations in the corpus

Nilprogressions and groups with moderate growth (2015, methodology): “Such a strong bound is currently only known for finite simple groups of bounded rank (see [9, 27, 42]).”
The structure of approximate groups (2011, result): “Various forms of the main theorem are also known in groups of Lie type of bounded dimension, as a consequence of results of many authors [5, 7, 8, 18, 19, 30, 31, 33, 43].”
Expander graphs, gonality, and variation of Galois representations (2010, methodology): “This was generalized to SL3 by Helfgott, to SLn (with some restriction) by Gill and Helfgott [29], and to all cases where G` = G(F`) for a simple split Chevalley group G/Z by Pyber-Szabó [53] and (independently) Breuillard-Green-Tao [10].”
Notes on Thin Matrix Groups (2012, background): “…call this “superstrong” approximation and it asserts that if we choose a finite symmetric (s ∈ S if s−1∈ S) generating set S of SLn(Z), then the congruence Cayley graphs (Xq , S) form an expander family as q goes to infinity (see [Hoory et al. 2006] for the definition and properties of expanders).”
Expansion in finite simple groups of Lie type (2013, methodology): “The general case is due independently to Pyber and Szabó [PS] and to the first, second and fourth authors [BGT3].”
On the diameter of permutation groups (2011, background): “Classical reasons aside, this will allow us to emphasise the link to [Hel08], [Hel11], [BGT11], [PS10], and [GH10], while repeating one of the main motifs: growth results from the action of a group on a set, often, as is the case here, by conjugation.”