Complexity in Bolza surface

Abstract

A surface in the Teichmüller space, where the systole function attains its maximum, is called a maximal surface. For genus two there exists a unique maximal surface which is called the Bolza surface. In this article, we study the complexity of the set of systolic geodesics on the Bolza surface. We show that any non-systolic geodesic intersects the systolic geodesics in $2n$ points, where $n\ge 5$. Furthermore, we show that there are $12$ second systolic geodesics on the Bolza surface and they form a triangulation of the surface.

Final version, to appear in the New York Journal of Mathematics