Closed geodesics on semi-arithmetic Riemann surfaces

Gregory Cosac; Cayo Dória

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arXiv · 2004.13683

Closed geodesics on semi-arithmetic Riemann surfaces

Gregory Cosac·Cayo Dória

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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 $g \geq 2,$ 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.

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BibTeX RIS

Gregory Cosac, Cayo Dória. 2020-04-28. Closed geodesics on semi-arithmetic Riemann surfaces. https://arxiv.org/abs/2004.13683

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