Locally symmetric space with and without short closed geodesics

Locally symmetric spaces arise in several areas of mathematics and are among the most structured spaces, exhibiting special harmony between geometry, analysis, arithmetic, algebra, and topology. Building on his celebrated arithmeticity theorem, Margulis conjectured that torsion-free cocompact arithmetic lattices of semisimple Lie groups are uniformly discrete. Geometrically, this means a uniform lower bound on the lengths of all closed geodesics for arithmetic locally symmetric spaces. This conjecture is widely open, even in the simplest case of compact arithmetic hyperbolic surfaces obtained as quotients of the hyperbolic plane. In joint work with M. Fraczyk, we proved that this is enough to prove Margulis' conjecture for all locally symmetric spaces of higher rank simple Lie groups. In fact, we establish uniform lower bounds on the lengths of ``most'' closed geodesics and prove that the main difficulty lies in rank one. The proof exploits in an essential way the arithmetic structure of these spaces.