SEMINAR: Lectures on Growth in Groups and Graphs Part II

Nick Gill (Bristol)

will continue his series of lectures on growth in groups and graphs

at the following times

Wednesday 21/7 at 11am

Friday 23/7 at 11am

and Tuesday 27/7 at 1pm

All seminars will be held in MLR2

I: Growth in soluble subgroups of GL_r(p)

Abstract: We show how to reduce the study of exponential growth in soluble subgroups of GL_r(p) to the nilpotent setting. We make use of ideas based on the sum-product phenomenon, as well as some machinery from linear algebraic groups. We will not assume any background from these areas.

II: An introduction to expanders

Abstract: This is a background seminar preparing the way for Seminar III, where we connect results on growth in simple groups to the explicit construction of families of expanders. In this seminar we will define what we mean by a family of expanders, stating (and sometimes even proving!!) background results that will be important later.

We will also try to explain why expanders are of such interest to so many different groups of people.

III: Using growth results to explicitly construct expanders

Abstract: We outline the method of Bourgain and Gamburd. They were the first to use results concerning growth in simple groups to explicitly construct expander graphs. Let S be a set in SL_2(Z) and define S_p to be the natural projection of S modulo p. Now let G_p be the Cayley graph of SL_2(F_p) with respect to the set S_p. Bourgain and Gamburd give precise results as to when the set of graphs {G_p : p a prime} forms a family of expanders. They make crucial use of the result of Helfgott (encountered in an earlier seminar) which states that "all generating sets in SL_2(p) grow".

All welcome