tailieunhanh - Đề tài " Growth and generation in SL2(Z/pZ) "

We show that every subset of SL2 (Z/pZ) grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators A of SL2 (Z/pZ), every element of SL2 (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. 1. Introduction . Background. Let G be a finite group. Let A ⊂ G be a set of generators of G. By definition, every g ∈ G can be expressed as a product of elements of. | Annals of Mathematics Growth and generation in SL2 Z pZ By H. A. Helfgott Annals of Mathematics 167 2008 601 623 Growth and generation in SL2 Z pZ By H. A. Helfgott Abstract We show that every subset of SL2 Z pZ grows rapidly when it acts on itself by the group operation. It follows readily that for every set of generators A of SL2 Z pZ every element of SL2 Z pZ can be expressed as a product of at most ỡ logp c elements of A u A-1 where c and the implied constant are absolute. 1. Introduction . Background. Let G be a finite group. Let A c G be a set of generators of G. By definition every g G G can be expressed as a product of elements of A u A-1. We would like to know the length of the longest product that might be needed in other words we wish to bound from above the diameter diam r G A of the Cayley graph of G with respect to A. The Cayley graph r G A is the graph V E with vertex set V G and edge set E ag g g G G a G A . The diameter of a graph X V E is maxV1 v2ey d v1 v2 where d v1 v2 is the length of the shortest path between v1 and v2 in X. If G is abelian the diameter can be very large if G is cyclic of order 2n 1 and g is any generator of G then gn cannot be expressed as a product of length less than n on the elements of g g-1 . However if G is non-abelian and simple the diameter is believed to be quite small Conjecture Babai BS . For every non-abelian finite simple group G diam r G A log G c where c is some absolute constant and G is the number of elements of G. This conjecture is far from being proved. Even for the basic cases viz. G An and G PSL2 Z pZ the conjecture has remained open until now these two choices of G seem to present already many of the main difficulties of the general case. The author was supported by a fellowship from the Centre de Recherches Mathematiques at Montreal. Travel was partially funded by the Clay Mathematics Institute. 602 H. A. HELFGOTT Work on both kinds of groups long predates the general conjecture in BS . Let us .