Marked Length Spectrum Rigidity for Surface Amalgams

In this paper, we show that simple, thick negatively curved two-dimensional P-manifolds, a large class of surface amalgams, are marked length spectrum rigid. That is, if two piecewise negatively curved Riemannian metrics (satisfying certain smoothness conditions) on a simple, thick two-dimensional P-manifold assign the same lengths to all closed geodesics, then they differ by an isometry up to isotopy. Our main theorem is a natural generalization of Croke and Otal’s celebrated results about marked length spectrum rigidity of negatively curved surfaces.