Two Applications of the Arithmetic Margulis Lemma

This course has two parts. In the first we will define a general notion of integer point in a higher Teichmüller space and two invariants thereof, the arithmetic dimension and the arithmetic height. The aim is to show finiteness, modulo the mapping class group action, of the number of integer points for which both invariants are bounded.
In the second part we will give the main steps leading to the recent result of Frączyk, Hurtado and Raimbault, stating that the topological complexity of an arithmetic locally symmetric space is bounded by its volume.
Both parts rely on the arithmetic Margulis lemma proven by the same authors as a consequence of Breuillard’s height gap theorem.