Bolza-like surfaces in the Thurston set

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 triangle decomposition of the surface. We define a surface as Bolza-like if its systolic geodesics decompose the surface into ( p , q , r ) (p, q, r) -triangles for some integers p , q , r p,q,r . In this article, we construct a one-parameter family of hyperbolic surfaces within the Thurston set χ g \chi _g , which, in turn, yields Bolza-like surfaces for infinitely many genera g ≥ 9 g\geq 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.