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A fairly detailed construction of arithmetic triangle groups can be found in Takeuchi's paper: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1240433796, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106.

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A fairly detailed construction of arithmetic triangle groups can be found in Takeuchi's paper: http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1240433796