We show that every subset of SL 2 (Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL 2 (Z/pZ), every element of SL 2 (Z/pZ) can be expressed as a product of at most O((log p) c ) elements of A ∪ A −1 , where c and the implied constant are absolute.

Abstract. Let G = SL 3 (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− for some > 0. Then |A · A · A| > |A| 1+δ , where δ > 0 depends only on .We will also study subsets A ⊂ G that do not generate G. Other results on growth and generation follow.

Abstract. Given a finite group G and a set A of generators, the diameter diam(Γ(G, A)) of the Cayley graph Γ(G, A) is the smallest ℓ such that every element of G can be expressed as a word of length at most ℓ in A ∪ A −1 . We are concerned with bounding diam(G) := maxA diam(Γ(G, A)).It has long been conjectured that the diameter of the symmetric group of degree n is polynomially bounded in n, but the best previously known upper bound was exponential in √ n log n. We give a quasipolynomial upper bound, namely, diam(G) = exp O((log n) 4 log log n) = exp (log log |G|)O (1) for G = Sym(n) or G = Alt(n), where the implied constants are absolute. This addresses a key open case of Babai's conjecture on diameters of simple groups. By a result of Babai and Seress (1992), our bound also implies a quasipolynomial upper bound on the diameter of all transitive permutation groups of degree n.

We give new bounds for the number of integral points on elliptic curves. The method may be said to interpolate between approaches via diophantine techniques and methods based on quasi-orthogonality in the Mordell-Weil lattice. We apply our results to break previous bounds on the number of elliptic curves of given conductor and the size of the 3 3 -torsion part of the class group of a quadratic field. The same ideas can be used to count rational points on curves of higher genus.

Abstract. This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL 2 (Z/pZ) as the basic example, as well as permutation groups. The emphasis will lie on the ideas behind the methods.