Stellar foliation structures on surfaces
Abstract
We introduce the notion of a zebra structure on a surface, which is a more general geometric structure than a translation structure or a dilation structure that still gives a directional foliation of every slope. We are concerned with the question of when a free homotopy class of loops (or a homotopy class of arcs relative to endpoints) has a canonical representative or family of representatives, either as closed leaves or chains of leaves joining singularities. We prove that such representations exist if the surface has a triangulation with edges joining singularities (in the zebra structure sense). In the special case when the surface is closed, we describe several geometric conditions that are equivalent to the existence of canonical representations in every homotopy class of closed curves.
Cite
@article{arxiv.2301.03727,
title = {Stellar foliation structures on surfaces},
author = {W. Patrick Hooper and Ferrán Valdez and Barak Weiss},
journal= {arXiv preprint arXiv:2301.03727},
year = {2023}
}
Comments
This version now proves Conjecture 1.10 from the first version as Theorem 1.6 in the new version; In the closed surface case, this proves several statements are equivalent to the existence of a closed trail in every free PRU homotopy class. 87 pages, 30 figures