Modul: MAT760 Ergodic Theory and Dynamical Systems Seminar

## Approximate lattices in higher rank simple Lie groups

Talk by Simon Machado

Speaker invited by: Prof. Dr. Corinna Ulcigrai

**Date:** 25.10.21 **Time:** 15.00 - 16.00 **Room:** Y27H28

Strong approximate lattices are defined as discrete approximate subgroups of finite co-volume in locally compact groups. To make sense of the notion of “finite co-volume” for a subset that is not a subgroup we need to introduce the notion of invariant hull. Given a closed subset X of a locally compact group G, the invariant hull of X is defined as the set of those subsets of G that cannot be distinguished locally from a translate of X. I will discuss certain properties of the invariant hull, and of approximate lattices in general. I will then explain how a careful analysis of Borel cocycles on the invariant hull, combined with cocycle superrigidity results, enables us to characterise the strong approximate lattices of SL_n(R) with n at least 3. They correspond to sets of matrices with coefficients in the set of Pisot-Vijayaraghavan numbers of some real number field K.