Superrigidity and arithmeticity for some aperiodic subsets in higher-rank simple Lie groups

Abstract: Meyer sets are fascinating objects: they are aperiodic subsets of Euclidean spaces that nonetheless exhibit long-range aperiodic order. Sets of vertices of the Penrose tiling (P3) and Pisot-Vijarayaghavan numbers of a real number field are some of the most well-known examples. In his pioneering work, Meyer provided a powerful and elegant characterisation of Meyer sets. Years later, Lagarias proved a similar characterisation starting from what seemed to be considerably weaker assumptions. A fascinating question asks whether Meyer’s and Lagarias’ results may be extended to more general ambient groups. In fact, a first result in that direction was already obtained in Meyer’s work: he proved a sum-product phenomenon which, implicitly, boiled down to a classification of Meyer sets in the group of affine transformations of the line. I will talk about a generalisation of both Meyer’s and Lagarias’ theorems to discrete subsets of higher-rank simple Lie groups. I will explain how this result can be seen as a generalisation of Margulis’ arithmeticity theorem and how it can be deduced from Zimmer’s cocycle superrigidity. We will see that, surprisingly, Pisot-Vijarayaghavan numbers appear naturally in this context too.

dynamical systemsnumber theory

Audience: general audience

New England Dynamics and Number Theory Seminar