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I am wondering whether every Fuchsian group (of a particular subclass, see below) is a finite index subgroup in some triangle group.

So, does there exists a cofinite non-uniform Fuchsian group which is NOT a subgroup of any triangle group? If so, can one write down an explicit example (via providing a fundamental domain or a set of generators or the like)?

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There is continuum of Fuchsian groups, while there are only countable many triangle groups (up to conjugation). –  Misha Oct 15 '12 at 13:21
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I don't know if this is explicit enough for you, but if you like Teichmuller theory and you think that translation surfaces are concrete objects, then you might like to check Corollary 9.6 of this nice article of C. McMullen http://www.ams.org/mathscinet-getitem?mr=1992827 where he constructs infinitely many cofinite non-uniform Fuchsian groups that are not commensurable to triangle groups simply by noticing that the trace fields of his examples lead to infinitely many quadratic fields while there are only finitely many quadratic fields coming from trace fields of triangle groups.

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