Your group is the fundamental group of a 2-dimensional orbifold with underlying surface the 2-sphere and 3 cone points of order p. It follows that it acts properly discontinuously and cocompactly by isometries on the 2-sphere when p=2, the Euclidean plane when p=3, and the hyperbolic plane when p>3. Hence it's infinite when p>2.
UPDATE:
Pete Clark's answer very eloquently explains the details that I outlined. I'd just like to add a couple of further remarks.
- To determine which sort of geometry (spherical, Euclidean or hyperbolic) an orbifold $O$ admits (ie upon which space your group acts as a discrete group if isometries) you just need to look at the (orbifold) Euler characteristic, defined to be
$\chi(O)=\chi(|O|)+\sum_i (1-1/p_i)$
where $|O|$ is the underlying surface and $p_i$ are the orders of your cone points. So we see that this is positive when $p=2$, zero when $p=3$ and negative when $p<3$.
- For a nice introduction to 2-dimensional orbifolds, I recommend Peter Scott's article The geometries of 3-manifolds.
