"Boundary slopes for the Markov ordering on relatively prime integer pairs"
Gaster, JonahA rational number p/q determines a simple closed curve on a once-punctured torus. When the torus is endowed with a complete hyperbolic metric, each rational gets a well-defined length. If the metric is chosen so that the torus is “modular” (that is, when its holonomy group is conjugate into PSL(2,Z)), the lengths of the curves have special arithmetic significance with connections to Diophantine approximation and number theory. Taking inspiration from McShane’s elegant proof of Aigner’s conjectures, concerning the (partial) ordering of the rationals induced by hyperbolic length on the modular torus, I will describe how hyperbolic geometry can be used to characterize monotonicity of the order so obtained along lines of varying slope in the $(q,p)$-plane.