### Diophantine Properties of Nilpotent Lie Groups

## Abstract

A finite set of elements in a connected real Lie group is "Diophantine" if non-identity short words in the set all lie far away from the identity. It has long been understood that in abelian groups, such sets are abundant. In this talk I will discuss recent work of Aka; Breuillard; Rosenzweig and de Saxce concerning this phenomenon (and its limitations) in the more general setting of nilpotent groups.