 Here, a geometric action is a cocompact, properly discontinuous action by isometries. This conjecture was partially solved by Grigori Perelman in his proof of the Geometrization conjecture, which states (in part) than any Gromov hyperbolic group that is a 3-manifold group must act geometrically on hyperbolic 3-space. However, it still remains to show that a Gromov hyperbolic group with a 2-sphere at infinity is a 3-manifold group.
