TITLE: For which triples $(i,j,k)$ is the presented group finite?
QUESTION [7 upvotes]: I was wondering when is the group $$\langle a,b \mid a^i, b^j, (ab)^k \rangle$$ finite?

Here are some examples:
Tetrahedral, Octahedral and Icosahedral groups: $\langle s,t \mid s^2, t^3, (st)^3 \rangle\,\!$, $\langle s,t \mid s^2, t^3, (st)^4 \rangle\,\!$, $\langle s,t \mid s^2, t^3, (st)^5 \rangle\,\!$
$D_{2n}$: $\langle r, f| r^n, f^2, (rf)^2\rangle$

https://en.wikipedia.org/wiki/Presentation_of_a_group

REPLY [7 votes]: The group presentation you have given is a presentation for a collection of groups called ordinary triangle groups or von Dyck groups. These are subgroups of larger general triangle groups.
The behavior is very different depending on whether $\frac 1i + \frac 1j + \frac 1k <1$, or $=1$, or $> 1$. These yield Hyperbolic, Euclidean, or Spherical triangle groups (respectively). Of these, only the spherical case corresponds to finite triangle groups and (correspondingly) finite von Dyck groups.
So in all, it will be finite iff $\frac 1i + \frac 1j + \frac 1k > 1$.