Work

Marked length spectrum rigidity in groups with contracting elements

Year: 2025

Type: article

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 and with contracting property, we prove that if the two actions have the same marked length spectrum, then the orbit map 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.

Source: Journal of the London Mathematical Society

DOI: 10.1112/jlms.70146

Authors: Renxing Wan, Xiaoyu Xu, Wenyuan Yang

Institutions: East China Normal University, Peking University

Language: English

Topic: Geometric and Algebraic Topology

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Natural Science Foundation of China, Science and Technology Commission of Shanghai Municipality, National Key Research and Development Program of China

Awards: G1609915050, G217537675, G7483175396, G8358677156, G8666907582