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On the arithmetic dimension of triangle groups


Authors: Steve Nugent and John Voight
Journal: Math. Comp. 86 (2017), 1979-2004
MSC (2010): Primary 11F03, 20H10; Secondary 11R52, 14G35
DOI: https://doi.org/10.1090/mcom/3147
Published electronically: November 8, 2016
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Abstract: Let $ \Delta =\Delta (a,b,c)$ be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles $ \pi /a,\pi /b,$
$ \pi /c$ drawn on the hyperbolic plane. We define the arithmetic dimension of $ \Delta $ to be the number of split real places of the quaternion algebra generated by $ \Delta $ over its (totally real) invariant trace field. Takeuchi has determined explicitly all triples $ (a,b,c)$ with arithmetic dimension $ 1$, corresponding to the arithmetic triangle groups. We show more generally that the number of triples with fixed arithmetic dimension is finite, and we present an efficient algorithm to completely enumerate the list of triples of bounded arithmetic dimension.


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Additional Information

Steve Nugent
Affiliation: Lady Margaret Hall College, Norham Gardens, Oxford OX2 6QA, United Kingdom
Email: steve.nugent@me.com

John Voight
Affiliation: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
Email: jvoight@gmail.com

DOI: https://doi.org/10.1090/mcom/3147
Received by editor(s): November 7, 2015
Received by editor(s) in revised form: January 26, 2016
Published electronically: November 8, 2016
Article copyright: © Copyright 2016 American Mathematical Society