Triangle group (awaiting review)

Given positive integers $l, m, n \ge 2$, the triangle group $\Delta(l, m, n)$ is given by the presentation $\langle a, b, c | a^2 = b^2 = c^2 = (ab)^l = (bc)^m = (ca)^n \rangle$. It is a Coxeter group, and forms the symmetry group of a tiling of either the sphere (if $\frac1l + \frac1m + \frac1n > 1$), the Euclidean plane (if $\frac1l + \frac1m + \frac1n = 1$) or the hyperbolic plane (if $\frac1l + \frac1m + \frac1n < 1$) by triangles with angles $\frac{\pi}l, \frac{\pi}m$ and $\frac{\pi}n$.

The Von Dyck group $D(l, m, n)$ is the subgroup of index $2$ generated by $x = ab, y = bc$ and $z = ca$. It has presentation $\langle x, y | x^l = y^m = (xy)^n \rangle$ and can be interpreted as the orientation preserving elements of the triangle group.