Marked Length Spectrum Rigidity in Groups with Contracting Elements

In this talk, we shall discuss the marked length spectrum rigidity problem in Riemannian geometry from a coarse geometric point of view. Namely, given two cobounded isometric actions of a group G on metric spaces X and Y, the coarse marked length spectrum rigidity asks whether the same length spectrum for the two actions ensures an almost G-equivariant isometry between X and Y. In contrast to the very few known classical rigidity results, we shall establish the coarse version, in a very general setting, for any group actions with contracting property. In addition, we prove a finer marked length spectrum rigidity from confined subgroups and further, geometrically dense subgroups. Our proof is based on the Extension Lemma and uses purely elementary metric geometry. This is joint work with Renxing Wan (ECNU) and Xiaoyu Xu (PKU).