Harald Helfgott

Growth in groups - abelian and otherwise

Take a set A. Consider A+A = {x+y:x,y in A}. How large
is A+A, compared to A? It could be large, or it could be small;
what does the answer tell us about A?

People were poking around abelian groups asking this sort of
questions: that is called "additive combinatorics" - a Hungarian
specialty now appreciated worldwide. What if the group is not abelian?
It turns out that some of the techniques and result developed for the
abelian case carry over. However, non-abelian groups have a face all
their own.

As it turns out, subsets A of "very non-abelian" groups - read:
simple, non-abelian groups, such as most classical matrix groups -
seem to grow rapidly, essentially without exceptions. By rapid growth
the following is meant: |A...A...A|>|A|^{1+epsilon},
epsilon>0 absolute.

We shall look at a proof for SL_2 and SL_3, and
perhaps glance at how things are shaping up for other matrix groups;
their structure as groups of Lie type is central to the matter.

Applications to diameters, expander graphs and the like keep popping
up. It happens that ``growth'' leads to ``growth'', after all.