Bolza-like surfaces in the Thurston set

TL;DR

This work studies Bolza-like surfaces in the Thurston set of Teichmüller space, extending maximal-systole phenomena beyond genus $2$. The authors construct Bolza-like surfaces for infinitely many genera $g$ (specifically $g=kn+1$, $k\ge 2$, $n\ge 4$, with $g\ge 9$) by tiling a torus with $mn$ copies of a hyperbolic square and lifting via an involution, producing a family $S_g(\epsilon)$ with monotone systole and a Bolza-like decomposition at $\epsilon=\frac{\pi}{12}$. At this critical parameter, the surface has $6g-6$ systolic geodesics that decompose the surface into $(p,q,r)$-triangles, establishing Bolza-like geometry. The paper then applies a Schmutz-style hexagon replacement to realize global maximal surfaces in genus $g$ (and punctured variants), and proves a symmetry result: any simple closed geodesic intersects the systolic set an even number of times. These results advance understanding of the Thurston spine and provide explicit infinite families of global maximal and symmetric systolic configurations in higher genus.

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 $g\geq 9$. 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.

Figures (10)

• Figure 2.1: Hyperbolic triangles
• Figure 3.1: $\triangle ABD$ and $\triangle BDC$ are the isometric triangles with interior angles at each of the vertices $A$ and $C$ is $\frac{\pi}{4}-\epsilon$.
• Figure 3.2:
• Figure 3.3:
• Figure 3.4: The torus $T_\epsilon$

Theorems & Definitions (9)

• Definition 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Theorem 3.4
• proof
• Theorem 5.1
• proof