Marked Length Spectrum and Arithmeticity

This dissertation considers the marked length spectrum of negatively curved closed Rieman- nian manifolds. The marked length spectrum is conjectured to be a complete invariant of the isometric type. We have yet to provide new results in this direction. But instead, show some improvements in the particular case of arithmetic manifolds. Namely, the set of solutions of ℓ(γ) = ℓ(η) is enough to pin down the metric up to homothety. This is the main result in (1). On the other hand, a similar idea leads to a new characterization of arithmetic metrics in the coarsely-geometric sense. The arithmetic metrics are exactly those metrics with infinitely many nontrivial commensurators.