Marked length spectra of Gromov hyperbolic space

Abstract

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

arXiv commentWe change the statement of the early version for readability