Orateur
Description
We introduce a new class of geometric structures on surfaces, called zebra structures, which generalize translation and dilation structures yet still induce directional foliations for every slope. Our primary goal is to determine when a free homotopy class of loops (or a homotopy class of arcs with fixed endpoints) admits a canonical representative—or a canonical family of representatives—realized as closed leaves or as chains of leaves connecting singularities. These canonical representatives are analogs of geodesic representatives in translation surfaces.
Our main result shows that representatives always exist provided the surface admits a triangulation whose edges connect the singularities in the sense of the zebra structure. In the special case where the surface is closed, we further characterize several geometric conditions under which every homotopy class of closed curves has a canonical representative.
This is joint work with P. Hooper and B. Weiss. Reference: https://arxiv.org/abs/2301.03727
