Abstract: This talk will is about work with Ben McReynolds and Matt Stover on variations of the higher Lehmer conjecture of F. Rodriguez Villegas. These conjectures concern the unit groups of algebraic number fields. I will make a connection between these conjectures and the Cheeger constants of certain flat manifolds which are naturally related to units. Cheeger constants measure the size of the minimal codimension one submanifold which cuts the original manifold into two disjoint pieces of equal volume.