  We prove that PSL(2,Z[1/p]) gives the first example of groups which are not
quasi-isometric to each other but have the same quasi-isometry group. Namely,
PSL(2,Z[1/p]) and PSL(2,Z[1/q]) are not quasi-isometric unless p=q, and,
independent of p, the quasi-isometry group of PSL(2,Z[1/p]) is PSL(2,Q). In
addition, we characterize PSL(2,Z[1/p]) uniquely among all finitely generated
groups by its quasi-isometry type.
