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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Geometry and Markoff's spectrum for $ \mathbb{Q}(i)$, I


Authors: Ryuji Abe and Iain R. Aitchison
Journal: Trans. Amer. Math. Soc. 365 (2013), 6065-6102
MSC (2010): Primary 57M50, 20H10, 53C22, 11J06
Published electronically: August 1, 2013
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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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Additional Information

Ryuji Abe
Affiliation: Department of Mathematics, Tokyo Polytechnic University, Atsughi, Kanagawa 243-0297, Japan
Email: ryu2abe@email.plala.or.jp

Iain R. Aitchison
Affiliation: Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
Email: I.Aitchison@ms.unimelb.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9947-2013-05850-3
PII: S 0002-9947(2013)05850-3
Received by editor(s): October 22, 2011
Received by editor(s) in revised form: March 30, 2012
Published electronically: August 1, 2013
Additional Notes: The first author was partially supported by Université de Tours (LMPT) and Université de Caen (LMNO)
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.