Boundary slopes for the Markov ordering on relatively prime pairs

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 $q\ge p\ge 0$. 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.

11 pages, 6 figures, comments welcome! v.2 corrects a mistake in section 4 of v.1; see Remark 4.2