Small Systle Sets and Coxeter Groups
Geometric Topology
2023-10-25 v1 Group Theory
Abstract
The systoles of a hyperbolic surface {\Sigma} are the shortest closed geodesics. We say that the systoles fill the surface if the set Syst({\Sigma}) of all systoles cuts {\Sigma} into polygons. We refine an idea of Schmutz [15] to construct closed hyperbolic surfaces {\Sigma} of arbitrarily large genus with a small set Syst({\Sigma}) that fills. In fact, for the surfaces {\Sigma} considered, the cardinality of Syst({\Sigma}) is in o(g/ ln g), where g is the genus of {\Sigma}. The proof is based on the theory Coxeter groups, combined with some elementary number theory.
Cite
@article{arxiv.2310.15531,
title = {Small Systle Sets and Coxeter Groups},
author = {Ingrid Irmer and Olivier Mathieu},
journal= {arXiv preprint arXiv:2310.15531},
year = {2023}
}