Marked length spectrum rigidity in groups with contracting elements

Abstract

This paper presents a study of the well-known marked length spectrum rigidity problem in the coarse-geometric setting. For any two (possibly non-proper) group actions $G\curvearrowright X_1$ and $G\curvearrowright X_2$ with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map $G{o}_1\to G{o}_2$ must be a rough isometry. In the special case of cusp-uniform actions, the rough isometry can be extended to the entire space. This generalises the existing results in hyperbolic groups and relatively hyperbolic groups. 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 study produces new results and recovers existing ones for many more interesting groups through a unified and elementary approach.