Some remarks on the geometry of arithmetic locally symmetric spaces

We discuss an arithmetic version of the classical (Zassenhaus or Margulis) lemma regarding the topology of the thin part of a locally symmetric space. We give several applications including bounds on the thin part of arithmetic locally symmetric spaces and some results regarding arithmetic groups generated by torsion elements. Based on works with Mikolaj Fraczyk and Jean Raimbault, and with David Fisher. All notions will be explained.