Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Marked length spectra of Gromov hyperbolic space

Yanlong Hao

Abstract

Let $(X,d)$, $(Y, d')$ be two roughly geodesically complete Gromov hyperbolic spaces under comparable isometric actions of $Γ$. Assume that the limit set $ΛΓ=\partial X\partial Y$. If spaces $X$ and $Y$ have the same asymptotic marked length spectrum, meaning that $$\lim_{{l_{d}([γ])\to \infty}}\frac{l_d(γ)}{l_{d'}(γ)}=1.$$ Then $(X,d)$ and $(Y,d')$ are $Γ$-equivariantly roughly isometric.

Marked length spectra of Gromov hyperbolic space

Abstract

Let , be two roughly geodesically complete Gromov hyperbolic spaces under comparable isometric actions of . Assume that the limit set . If spaces and have the same asymptotic marked length spectrum, meaning that Then and are -equivariantly roughly isometric.
Paper Structure (19 sections, 24 theorems, 98 equations)

This paper contains 19 sections, 24 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Spectrally rigid set
  3. Marked length spectrum rigidity
  4. Roughly geodesically complete hyperbolic space
  5. Gromov product and Gromov hyperbolic spaces
  6. Quasi-isometries, roughly geodesic space
  7. Busemann cocycle (rough Busemann cocycle)
  8. Barycenter method and relative Busemann function
  9. Some lemmas
  10. Main idea: length relation
  11. Proof of Theorem \ref{['mthm: sparse rigid set']} and \ref{['mthm: sparse multiplicative rigid set']}
  12. Proof of Theorem \ref{['mthm: sparse rigid set']}
  13. Proof of Theorem \ref{['mthm: sparse multiplicative rigid set']}
  14. Rough cross-ratio of roughly geodesic hyperbolic space and Busemann functions
  15. Infinite-barycenter map
  16. ...and 4 more sections

Key Result

Theorem A

Let $(X, d_1)$ and $(Y, d_2)$ be two Gromov hyperbolic spaces under non-elementary isometric actions by a group $\Gamma$. Assume that there is a $\Gamma$-equivalent continuous map from $\Lambda_X \Gamma$ to $\Lambda_Y \Gamma$, where $\Lambda_X \Gamma$ and $\Lambda_Y \Gamma$ are the limit sets of $\G then $\ell_{d_1}=\ell_{d_2}$ on $\Gamma$.

Theorems & Definitions (52)

  • Theorem A
  • Corollary 1.1
  • Theorem B
  • Remark 1.2
  • Theorem C
  • Theorem 1.3
  • Theorem D
  • Definition 1.4
  • Example 1.5
  • Example 1.6
  • ...and 42 more