 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 . 
