Boundary slopes for the Markov ordering on relatively prime pairs

TL;DR

This paper characterizes the boundary slopes where the Markov ordering on relatively prime pairs is monotone, using the stable norm on the modular torus's homology, confirming several conjectures in the process.

Contribution

It provides a precise characterization of monotone slopes in the Markov ordering, advancing understanding of the structure of relatively prime pairs.

Findings

Confirmed conjectures of Lee-Li-Rabideau-Schiffler.

Explicit computation of slopes at the corners of the stable norm ball.

Established conditions for monotonicity in the Markov ordering.

Abstract

Following McShane, we employ the stable norm on the homology of the modular torus to investigate the Markov ordering on the set of relatively prime integer pairs $(q,p)$ with $q\ge p\ge0$. Our main theorem is a characterization of slopes along which the Markov ordering is monotone with respect to $q$, confirming conjectures of Lee-Li-Rabideau-Schiffler that refine conjectures of Aigner. The main tool is an explicit computation of the slopes at the corners of the stable norm ball for the modular torus.