Closed geodesics on semi-arithmetic Riemann surfaces
Geometric Topology
2020-09-02 v2 Metric Geometry
Number Theory
Abstract
In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means of number theory and hyperbolic geometry. First, we show the existence of infinitely many semi-arithmetic Riemann surfaces of various shapes and prove that their systoles are dense in the positive real numbers. Furthermore, this leads to a construction, for each genus of infinite families of semi-arithmetic surfaces with pairwise distinct invariant trace fields, giving a negative answer to a conjecture of B. Jeon. Finally, for any semi-arithmetic surface we find a sequence of congruence coverings with logarithmic systolic growth and, for the special case of surfaces admitting modular embedding, we are able to exhibit explicit constants.
Cite
@article{arxiv.2004.13683,
title = {Closed geodesics on semi-arithmetic Riemann surfaces},
author = {Gregory Cosac and Cayo Dória},
journal= {arXiv preprint arXiv:2004.13683},
year = {2020}
}