SEMINAR: Lectures on Growth in Groups and Graphs

Nick Gill (University of Bristol) will be giving a series of three talks on the topic of Growth in Groups and Graphs as follows:

* Tuesday 4 May, 12 noon, MLR2, I: Sum-Product

* Friday 7 May, 11am, MLR3, II: Growth in Groups of Lie Type

* Friday 14th May 11am MLR3, III: Escape

These are part of a UWA research collaboration award funding visits to Perth by Nick Gill and Harald Helfgott. The aim of the talks is to give the necessary background and introduction to research in the area. Nick will give a further three talks later in the year. Nick has prepared a page of supporting material at http://www.maths.bris.ac.uk/~manpg/austlit.html

The titles and abstracts for the first three talks are as follows.

I: SUM-PRODUCT We introduce the idea of growth in groups, before focussing on the abelian setting. We take a first look at the sum-product principle, with a brief foray into the connection between sum-product results and incidence theorems.

We then focus on Helfgott’s restatement of the sum-product principle in terms of groups acting on groups.

II: GROWTH IN GROUPS OF LIE TYPE Since Helfgott first proved that “generating sets grow” in SL_2(p) and SL_3(p), our understanding of how to prove such results has developed a great deal. It is now possible to prove that generating sets grow in any finite group of Lie type; what is more the most recent proofs are very direct – they have no recourse to the incidence theorems of Helfgott’s original approach.

We give an overview of this new approach, which has come to be known as a ”pivotting argument”. There are five parts to this approach, and we outline how these fit together.

III: ESCAPE The principle of “escape from subvarieties” is the first step in proving growth in groups of Lie type. We give a proof of this result, and its most important application (for us) – the construction of regular semisimple elements.

We then examine other related ideas from algebraic geometry, in particular the idea of non-singularity.