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I'm seeking a reference or a sketch for any sort of normal form that would enable rapid enumeration without redundancies of the elements of hyperbolic triangle groups and/or von Dyck groups.

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  • $\begingroup$ might be here: ams.org/mathscinet-getitem?mr=1465327 $\endgroup$ – Ian Agol May 14 '12 at 16:06
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    $\begingroup$ @Ian: It is a very interesting paper. But doesn't it follow from the thesis I cited? That would be remarkable because the paper is very non-elementary (comparing to the thesis). Perhaps the main difference is that the thesis gives non-geodesic normal forms while the paper gives geodesic normal forms? Anyway, thanks for the reference. $\endgroup$ – user6976 May 14 '12 at 18:56
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This thesis contains a complete rewriting system for every triangle group (see Section C at the end of the thesis). Then normal forms are just words that do not contain left parts of the rewriting rules (which is just a finite set of words). This gives enumeration of all elements without redundancies.

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The general reference for this topic is "Word processing in groups", Epstein et. al., MR1161694 (93i:20036). Your groups all have automatic structures, and that book gives an enumeration method as you ask for, which applies to any automatic structure on a group.

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