Geometry and Markoff’s spectrum for ℚ(𝕚), I

Abstract

We develop a study of the relationship between geometry of geodesics and Markoff’s spectrum for Q ( i ) \mathbb {Q}(i) . There exists a particular immersed totally geodesic twice punctured torus in the Borromean rings complement, which is a double cover of the once punctured torus having Fricke coordinates ( 2 2 , 2 2 , 4 ) (2\sqrt {2}, 2\sqrt {2}, 4) . The set of the simple closed geodesics on this once punctured torus is decomposed into two subsets. The discrete part of Markoff’s spectrum for Q ( i ) \mathbb {Q}(i) (except for one) is given by the maximal Euclidean height of the lifts of the simple closed geodesics composing one of the subsets.