Two-color Markoff graph and minimal forms

Abstract

Diophantine equations [Formula: see text] and [Formula: see text] are connected with the arithmetic once punctured torus groups having Fricke coordinates [Formula: see text] and [Formula: see text]. We describe and analyze the structure of these equations by a graph, which we call a two-color Markoff graph. The faces of the graph are labeled by matrices in the corresponding once punctured torus group, the word representation of which is easily described owing to arithmetic properties of the graph. As an application, we show that these matrices define Markoff minimal forms attaining the values of the Markoff spectrum given by the positive integer solutions of these equations.