Journal of the European Mathematical Society


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Volume 13, Issue 3, 2011, pp. 761–851
DOI: 10.4171/JEMS/267

Growth in SL3(ℤ/pℤ)

H. A. Helfgott[1]

(1) School of Mathematics, University of Bristol, University Walk, BS8 1TW, BRISTOL, UNITED KINGDOM

Let G = SL3(ℤ/pℤ), p a prime. Let A be a set of generators of G. Then A grows under the group operation.

To be precise: denote by |S| the number of elements of a finite set S. Assume |A| < |G|1-ε for some ε > 0. Then |A ∙ A ∙ A| > |A|1+δ, where δ > 0 depends only on ε.

Keywords: Cayley graphs, finite groups, generation, diameter, expander graphs

Helfgott H. Growth in SL3(ℤ/pℤ). J. Eur. Math. Soc. 13 (2011), 761-851. doi: 10.4171/JEMS/267