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 $4$-space is $7$-dimensional. Among them, there is a $5$-dimensional parameter space of linked thrice-punctured sphere groups. In particular, there is a $1$-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.
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