Bolza-like surfaces in the Thurston set

Abstract

A surface in the Teichmüller space, where the systole function admits its maximum, is called a maximal surface. For genus two, a unique maximal surface exists, which is called the Bolza surface, whose systolic geodesics give a triangulation of the surface. We define a surface as Bolza-like if its systolic geodesics decompose the surface into $(p,q,r)$-triangles for some integers $p,q,r$. In this article, we will provide a construction of Bolza-like surfaces for infinitely many genera $g\ge 9$. Next, we see an intriguing application of Bolza-like surfaces. In particular, we construct global maximal surfaces using these Bolza-like surfaces. Furthermore, we study a symmetric property satisfied by the systolic geodesics of our Bolza-like surfaces. We show that any simple closed geodesic intersects the systolic geodesics at an even number of points.

Final version, Accepted for publication in the Proceedings of the American Mathematical Society