# For which triples $(i,j,k)$ is the presented group finite?

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: $\langle s,t \mid s^2, t^3, (st)^3 \rangle\,\!$, $\langle s,t \mid s^2, t^3, (st)^4 \rangle\,\!$, $\langle s,t \mid s^2, t^3, (st)^5 \rangle\,\!$

$D_{2n}$: $\langle r, f| r^n, f^2, (rf)^2\rangle$

• Is there a finite group presented this way with none of i,j,k equal to 2? – user58512 Feb 5 '13 at 20:26
• Finite if and only if 1/i + 1/j + 1/k is bigger than 1. Look up triangle groups. – user641 Feb 5 '13 at 20:29

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 $\frac 1i + \frac 1j + \frac 1k > 1$.