Approximate groups and applications to the growth of groups
Matthew Tointon
Tuesday, April 25, 2017, 10:50 – 12:00, Math -101
Abstract:
| Given a set A in a group, write $A^n$ for the set of all products $x_1…x_n$ with each $x_i$ belonging to A. Roughly speaking, a set A for which $A^2$ is “not much larger than” A is called an “approximate group”. The “growth” of A, on the other hand, refers to the behaviour of $ | A^n | $ as n tends to infinity. Both of these have been fruitful areas of study, with applications in various branches of mathematics. |
Remarkably, understanding the behaviour of approximate groups allows us to convert information $A^k$ for some single fixed k into information about the sequence $A^n$ as n tends to infinity, and in particular about the growth of A.
| In this talk I will present various results describing the algebraic structure of approximate groups, and then explain how to use these to prove new results about growth. In particular, I will describe work with Romain Tessera in which we show that if $ | A^k | $ is bounded by $Mk^D$ for some given M and D then, provided k is large enough, | A^n | is bounded by $M’n^D$ for every n > k, with M’ depending only on M and D. This verifies a conjecture of Itai Benjamini. |