"Diophantine Geometry and uniform growth of groups"

Abstract: A basic question in geometric group theory is to understand
the asymptotic behavior of the number of all products of n elements
chosen from a given subset of the group. Two famous results in this
direction are Gromov's theorem on groups with polynomial growth and
the Tits alternative regarding linear groups. In the talk I will
survey a number of recent developments in which new ideas from
additive combinatorics and arithmetic geometry have played a key role.