Marked length spectrum rigidity from rigidity on subsets
arXiv:2304.13209
Abstract
We introduce a new method for studying length spectrum rigidity problems based on a combination of ideas from dynamical systems and geometric group theory. This allows us to compare the marked length spectrum of metrics and distance-like functions coming from various geometric origins. Using our new perspective, we provide concise proofs of well-known length spectrum rigidity results and are able to extend classical results to a variety of new settings. Our methods rely on studying Manhattan curves and a coarse geometric analogue of Teichmüller space equipped with a symmetrized version of the Thurston metric.
26 pages (Version 2 is significantly different from Version 1: the final two sections and results regarding approximate rigidity have been removed from V1. These results will appear in a different work. New results and applications have also been added to V2.)