Growth in finite simple groups of Lie type of bounded rank

Abstract

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.

84 pages, various new results concerning non-growing subsets of linear groups were added

References in corpus (8)

• The Elementary Theory of the Frobenius Automorphisms
• Expander graphs, gonality and variation of Galois representations
• Expansion in $S{L}_d(O_K/I)$, $I$ square-free
• Growth in solvable subgroups of GL_r(Z/pZ)
• Approximate subgroups of linear groups
• Expansion properties of finite simple groups
• Growth in finite simple groups of Lie type
• Linear Approximate Groups

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• Splitting fields of characteristic polynomials of random elements in arithmetic groups
• Spectral gap in the group of affine transformations over prime fields
• A Product Theorem in simple Lie groups
• Sieve methods in group theory I: Powers in Linear groups
• Growth in groups: ideas and perspectives
• A nilpotent Freiman dimension lemma
• Mixing for progressions in non-abelian groups
• Using Expander Graphs to test whether samples are i.i.d