Given a closed manifold $M$ equipped with a Riemannian metric $g$ of negative sectional curvature, one can associate a positive real number $\ell_g(c)$ to each conjugacy class $c$ of the fundamental group $\Gamma=\pi_1(M)$, namely the $g$-length of the shortest loop on $M$ in the free homotopy class $c$.

In the talk we will discuss some conjectures and results, old and new, related to the function $\ell_g(c)$, arithmeticity of rank one lattices, and commensurators. 

In particular, It turns out that in the world of negatively curved manifolds the arithmetic locally symmetric ones can be detected using abstract commensurators and coarse-geometry.

The talk is based on a joint work with Yanlong Hao.