Thrice-punctured sphere groups in hyperbolic 4-space

Abstract

A thrice-punctured sphere group is a non-elementary group generated by two parabolic isometries whose product is a parabolic isometry. We prove that the deformation space of a thrice-punctured sphere group acting on hyperbolic 4 4 -space is 7 7 -dimensional. Among them, there is a 5 5 -dimensional parameter space of linked thrice-punctured sphere groups. In particular, there is a 1 1 -parameter family of discrete linked thrice-punctured sphere groups such that the rotation angles of the two parabolic generators and the product of the generators are fixed.