# How to find quotients of infinite triangle groups or von Dyck groups?

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.

• In particular, your group is finite precisely when $1/l+1/m+1/n>1$. – HJRW May 15 '13 at 11:22
• Ketan: Now it is no longer clear what your question is. Are you interested in finite subgroups of these groups? If so, they are all cyclic, whose generators are conjugate to powers of $S$ or $T$ or $ST$. What else do you need to know? (I do not know what "famous group" means.) – Misha May 15 '13 at 12:02