Marked length spectrum rigidity for relatively hyperbolic groups

TL;DR

This paper establishes a coarse length spectrum rigidity for relatively hyperbolic groups, showing that identical marked length spectra imply the metrics are close, thus extending Fujiwara's earlier results.

Contribution

It generalizes the marked length spectrum rigidity theorem to relatively hyperbolic groups, providing new insights into their geometric structure.

Findings

Rigidity theorem proven for relatively hyperbolic groups

Marked length spectrum determines metrics up to uniform closeness

Extends previous results by Fujiwara to a broader class of groups

Abstract

We consider a coarse version of the marked length spectrum rigidity: given a group with two left invariant metrics, if the marked length spectrum (the translation length function) under the two metrics are the same, then the two metrics are uniformly close. We prove the rigidity theorem for relatively hyperbolic groups. This generalizes a result of Fujiwara.