Closed geodesics on semi-arithmetic Riemann surfaces

In this article, we study geometric aspects of semi-arithmetic Riemann surfaces by means number theory and hyperbolic geometry. First, show the existence infinitely many various shapes prove that their systoles are dense in positive real numbers. Furthermore, leads to a construction, for each genus $g \geq 2,$ infinite families with pairwise distinct invariant trace fields, giving negative answer conjecture B. Jeon. Finally, any surface find sequence congruence coverings logarithmic systolic growth and, special case admitting modular embedding, able exhibit explicit constants.