Adam's blog http://cp4space.wordpress.com/ currently has "<s, t : s^10 = t^3 = (st)^2 = 1> This is a surprisingly exciting group. Firstly, it is straightforward to verify from its presentation that the group is a hyperbolic *Dyck group*, *i.e.* the orientation-preserving symmetries of the kaleidoscopic tiling of the hyperbolic plane by triangles with interior angles (π/2, π/3, π/10). [...] In other words, it is the group generated by the following Möbius transformations: - s(z) = z exp(i pi/5); - any t with the property that t(t(t(z))) = s(t(s(t(z)))) = z for all z. Now, this is a ring-theoretic definition, so we can apply a ring automorphism and replace exp(pi/5) with exp(3pi/5) in the above definition without changing the resulting abstract group. However, this is entirely different from a topological perspective, since we have a group of spherical symmetries — rotations — rather than hyperbolic symmetries. So we can regard this as a dense subgroup of SO(3) generated by two rotations." Question: Does this have a "group volume", and can it show the expected magnitude of an element of SO(3) is pi/2 + 2/pi? --rwg