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

Abe, Ryuji; Aitchison, Iain

doi:10.1090/S0002-9947-2013-05850-3

Geometry and Markoff’s spectrum for $\mathbb {Q}(i)$, I
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by Ryuji Abe and Iain R. Aitchison PDF

Trans. Amer. Math. Soc. 365 (2013), 6065-6102 Request permission

Abstract:

We develop a study of the relationship between geometry of geodesics and Markoff’s spectrum for $\mathbb {Q}(i)$. 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 $(2\sqrt {2}, 2\sqrt {2}, 4)$. 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 $\mathbb {Q}(i)$ (except for one) is given by the maximal Euclidean height of the lifts of the simple closed geodesics composing one of the subsets.

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