Approximate marked length spectrum rigidity in coarse geometry

TL;DR

This paper investigates how approximate comparisons of marked length spectra can determine geometric properties of group actions, extending rigidity results to broader contexts beyond hyperbolic groups.

Contribution

It introduces an approximate version of marked length spectrum rigidity applicable to various group actions, including non-hyperbolic groups, and refines existing results for negatively curved manifolds.

Findings

Supremum of quotient of marked length spectra is approximately determined by finite conjugacy classes.

Results extend to non-hyperbolic groups like mapping class groups.

Applicable to actions on Cayley graphs and curve graphs.

Abstract

We compare the marked length spectra of isometric actions of groups with non-positively curved features. Inspired by the recent works of Butt we study approximate versions of marked length spectrum rigidity. We show that for pairs of metrics, the supremum of the quotient of their marked length spectra is approximately determined by their marked length spectra restricted to an appropriate finite set of conjugacy classes. Applying this to fundamental groups of closed negatively curved Riemannian manifolds allows us to refine Butt's result. Our results however apply in greater generality and do not require the acting group to be hyperbolic. For example we are able to compare the marked length spectra associated to mapping class groups acting on their Cayley graphs or on the curve graph.