Some arithmetic aspects of ortho-integral surfaces

TL;DR

This work characterizes ortho-integral hyperbolic surfaces, where every orthogeodesic has integral cosh-length, and proves finiteness of such surfaces for fixed topology. It achieves complete classifications of OI pairs of pants and OI one-holed tori, and shows their doubles are arithmetic genus-2 surfaces coming from quaternion algebras over $\mathbb{Q}$, using a quadratic-form and Clifford-algebra framework. It further demonstrates the existence of infinitely many pairwise non-commensurable OI surfaces by gluing identical OI pants, providing a geometric mechanism for non-commensurability. The paper also develops a bridge between hyperbolic geometry and arithmetic via hexagon-based data, invariant trace fields, and Takeuchi’s criterion, yielding explicit arithmetic invariants for the doubled surfaces. Collectively, these results illuminate the arithmetic structure of OI surfaces and their moduli, and offer a concrete path to generate large families of non-commensurable examples.

Abstract

We investigate ortho-integral (OI) hyperbolic surfaces with totally geodesic boundaries, defined by the property that every orthogeodesic (i.e. a geodesic arc meeting the boundary perpendicularly at both endpoints) has an integer cosh-length. We prove that while only finitely many OI surfaces exist for any fixed topology, infinitely many commensurability classes arise as the topology varies. Moreover, we completely classify OI pants and OI one-holed tori, and show that their doubles are arithmetic surfaces of genus 2 derived from quaternion algebras over $\mathbb{Q}$.

Figures (4)

• Figure 1: Consider a pair of pants with a standard orthobasis with labels $\{P,B,R\}$ for pink, blue and red. The top four figures show four orthogeodesic arcs with cutting sequence $P$, $PR$, $PRPR$ and $PRBR$ starting from an interval $I$. The bottom figures show the orthotree parametrizing orthogeodesic starting from $I$.
• Figure 2: A pair of pants $P$ with the standard orthobasis $\mathcal{H} = \{\alpha,\beta,\gamma\}$ and three geodesic boundary components $\delta_{\alpha},\delta_{\beta},$ and $\delta_{\gamma}$.
• Figure 3: The all right-angled hexagon $H(a,b,c)$.
• Figure 4: Gluing copies of OI pair of pants $P(a,a,a)$ without twist, for a fixed $a\in \{2,3\}$.

Theorems & Definitions (43)

• Definition 1.1: doan2021
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 2.1: Extended version of Mumford's Compactness Criterion
• Theorem 2.2
• Theorem 2.3: doan2021, Theorem 1.2
• Remark 2.4
• Theorem 2.5: Bavard
• Remark 2.6