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Marked length spectrum rigidity in groups with contracting elements

Renxing Wan, Xiaoyu Xu, Wenyuan Yang

TL;DR

This work establishes a broad, elementary coarse-geometric MLS rigidity theory for group actions with contracting elements. Central to the framework is the Extension Lemma, which, together with simultaneously contracting elements, yields that equal marked length spectra force the orbit maps on orbits to be rough isometries, and in cusp-uniform contexts extend this rigidity to the ambient spaces. The results encompass hyperbolic and relatively hyperbolic groups, and extend to non-geodesic metrics like Green metrics. Subgroup- and subset-based rigidity are developed via confined/geometrically dense subgroups and Manhattan curves, respectively, yielding flexibility to derive MLS rigidity from a variety of substructures. Collectively, the paper provides a unified, elementary approach that recovers and extends MLS rigidity phenomena across several classical and new settings, including cusp-uniform actions, confinement, and growth-controlled subsets.

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 $Go_1\to Go_2$ must be a rough isometry. In the special case of cusp-uniform actions, the rough isometry can be extended to the entire space. This generalizes 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.

Marked length spectrum rigidity in groups with contracting elements

TL;DR

This work establishes a broad, elementary coarse-geometric MLS rigidity theory for group actions with contracting elements. Central to the framework is the Extension Lemma, which, together with simultaneously contracting elements, yields that equal marked length spectra force the orbit maps on orbits to be rough isometries, and in cusp-uniform contexts extend this rigidity to the ambient spaces. The results encompass hyperbolic and relatively hyperbolic groups, and extend to non-geodesic metrics like Green metrics. Subgroup- and subset-based rigidity are developed via confined/geometrically dense subgroups and Manhattan curves, respectively, yielding flexibility to derive MLS rigidity from a variety of substructures. Collectively, the paper provides a unified, elementary approach that recovers and extends MLS rigidity phenomena across several classical and new settings, including cusp-uniform actions, confinement, and growth-controlled subsets.

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 generalizes 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.
Paper Structure (32 sections, 82 theorems, 94 equations)

This paper contains 32 sections, 82 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background and motivations
  3. MLS rigidity for actions with contracting property
  4. MLS rigidity from subgroups
  5. MLS rigidity from subsets
  6. Key technique: Extension Lemma
  7. Preliminaries
  8. Notations and definitions
  9. Entry and exit points.
  10. Marked Length Spectrum.
  11. Contracting system
  12. Admissible path
  13. Group actions with contracting property
  14. Extension lemma
  15. Simultaneously contracting elements
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose that a group $G$ acts isometrically on two geodesic metric spaces $(X_1, d_1)$ and $(X_2, d_2)$ with contracting property (cf. DEF: Contracting Property), respectively. Then $G$ has MLS rigidity: if $\ell_{d_1}(g)=\ell_{d_2}(g)$ for every $g\in G$, then the orbit map $\rho: Go_1\to Go_2$ is

Theorems & Definitions (186)

  • Definition 1.1
  • Remark 1.1
  • Theorem 1.1: \ref{['THM: Main Rigidity']}
  • Theorem 1.2: \ref{['COR: Rigidity from SC']}
  • Theorem 1.3: \ref{['THM: Cusp Uniform']}
  • Corollary 1.1: \ref{['MLSGreenMetrics']}
  • Theorem 1.4: \ref{['THM: Normal Subgroup Rigidity']}
  • Corollary 1.2: \ref{['COR: Sequence of Normal Subgroup']}
  • Theorem 1.5: \ref{['Thm: MLS from confined subgps']}
  • Theorem 1.6: \ref{['THM: MLSR from Geometrically Dense Subgroup']}
  • ...and 176 more