UBC Mathematics Colloquium

## Harald Helfgott

(University of Bristol)

## Fri., March 5, 2010, 3:00pm, MATX 1100

Abstract:

Let A be a finite subset of a group G. How rapidly does A grow?

More precisely: let |S| be the number of elements of a finite set S. In 2005, I proved that, for G = SL_2(Z/pZ), p a prime, A\subset G such that A generates G and
|A|<=|G|^{1-epsilon}, epsilon>0, we have

|A A A| >> |A|^{1+delta},
(*)

where A A A = {x y z: x,y,z\in A}, and delta>0 and the implied constant depend only on epsilon.

This implies directly that the diameter of any Cayley graph of G is polylogarithmic (Babai's conjecture). Further implications on expander graphs were derived by Bourgain and Gamburd (and used by Bourgain, Gamburd and Sarnak in their work on the {\em affine sieve}).

In 2008, I proved the same result for SL_3(Z/pZ). Half of the proof had become fully general in the process, but much work remained to be done. Nick Gill and I extended the result to small subsets of SL_n in 2009.

This January, two different teams (Pyber and Szabo; Breuillard, Green and Tao) announced proofs of (*) valid for all finite simple groups of Lie type. Their success is based in part on a strengthening of some of my intermediate results from my paper in SL_3, apparently inspired by papers by Larsen-Pink and Hrushovski-Pillay. The process of making ideas of growth ("pivoting" or "bootstrapping") independent from the context of the sum-product theorem has also reached its natural conclusion.