Closed geodesics on semi-arithmetic Riemann surfaces

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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The first author was supported by CNPq-Brazil research grant 141204/2016-8.

Dória is grateful for the support of FAPESP grant 2018/15750-9.

Received 30 August 2020

Received revised 3 June 2021

Accepted 26 October 2021

Published 23 February 2023