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](https://mathoverflow.net/questions/118035/order-of-elements) 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](https://mathoverflow.net/questions/118035/order-of-elements), 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. 



Finite Quotients of Free Groups
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. 


