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Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of places including the book by Maclachlan and Reid, p. 159-160, where one finds an explicit presentation in terms of a suitable quaternion algebra; and in papers by Elkies and others where the appropriate order is specified, as well, and congruence subgroups are studied. Are there any texts where other triangle groups are dealt with in detail, in particular with regard to congruence subgroups and the corresponding Riemann surfaces? I am particularly interested in the (2,3,8) case and the closely related (3,3,4).

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Looks like Takeuchi gives more info. in another paper: – Ian Agol Oct 27 '13 at 3:08
Thanks for the link. – katz Oct 27 '13 at 14:38

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