Question

It is interesting FACT that given $l,m,n\geq 2$, there is (are) a finite group with elements $a,b$ such that $o(a)=l, o(b)=m$, and $o(ab)=n$ (see link for a nice example by Derek Holt / B. Sury).

Although, the group

$G_{l,m,n}=\langle a,b\colon a^l,b^m,(ab)^n\rangle$

contains elements with above property (?), it need not be finite. But for certain values of $l,m,n$, $G_{l,m,n}$ is finite (and these are interesting groups by their geometry):

$G_{2,2,n}\cong D_{2n}$,

$G_{2,3,3}\cong A_4$,

$G_{2,3,4}\cong S_4$,

$G_{2,3,5}\cong A_5$.

A natural question I would like to ask is:

For what values of $l,m,n$, the group $G_{l,m,n}$ is finite?

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I wondered by the above FACT, and very useful discussion on it in mathoverflow, with nice answer by Derek Holt/ B. Sury. I tried to solve the question in the link, and for $(l,m,n)=(2,2,n)$, I quickly found that the group $D_{2n}$ is the best example for it. Then I came up with the natural question above. This question may have been discussed with different point of view; but I didn't know too much about it.

Also, if possible, one may suggest(and edit) suitable title for this question.

Answer 1

These groups are called von Dyck groups, see e.g. here. Von Dyck group $D(l,m,n)$ is finite if and only if it is of spherical type: $$ \chi=-1+ l^{-1} + m^{-1} + n^{-1}>0. $$ A side note: Von Dyck groups are fundamental groups of 2-dimensional oriented orbifolds. The number $\chi$ above is the orbifold Euler characteristic. This is explained nicely, for instance, in Peter Scott's paper "Geometries of 3-manifolds", Bull. London Math. Soc. 15 (1983), no. 5, 401–487.