**Abstract:** The standard Hopf map from the 3-sphere to the 2-sphere has long been used to get topological information, specifically, homotopy groups of spheres. Less used is the rich geometry associated with the Hopf map -- in particular, the wealth of Clifford torii given by taking the inverse images of circles in the 2-sphere. The fact that they are of constant mean curvature is easily proved by using the relation of the group structure on the 3-sphere to the Hopf map. But there are some other 3-dimensional spaces with group stuctures, the simplest of which is Euclidean space, but more interestingly, the anti-de Sitter Lorentz space. Treating them analogously leads to some other Hopf maps, with consequent families of cylinders, which are constant-mean curvature homogeneous intrinsically- flat surfaces.