I need the following information about the quotients of infinite triangle (or von Dyck) groups.

(1) Let $G(l,m,n)$ defined as $S^l$ =$T^m$ = $(ST)^n$ = $E$ is the hyperbolic ($1/l+1/m+1/n<1$) triangle group. What are the additional conditions $S$ and $T$ should satisfy in order to ensure that the given quotients of this group is finite?

(2) Is there exist any reference where such quotient groups have been identified as isomorphic to other famous group?

I am a student of Physics and have very limited knowledge of the group theory, so simple answer will be very helpful.

Thanks in advance.

Ketan Patel

quotientsrather thansubgroups. So your question becomes: what relations must I add to obtain a finite group? (It's a question with no very straight-forward answer by the way.) $\endgroup$ – Nick Gill May 15 '13 at 12:34quotients, not finitesubgroups. $\endgroup$ – HJRW May 15 '13 at 13:02