Thrice-punctured spheres in hyperbolic $3$-manifolds

Abstract

The work of W. Thurston has stimulated much interest in the volumes of hyperbolic 3-manifolds. In this paper, it is demonstrated that a 3-manifold M' obtained by cutting open an oriented finite volume hyperbolic 3-manifold M along an incompressible thrice-punctured sphere S and then reidentifying the two copies of S by any orientation-preserving homeomorphism of S will also be a hyperbolic 3-manifold with the same hyperbolic volume as M. It follows that an oriented finite volume hyperbolic 3-manifold containing an incompressible thrice-punctured sphere shares its volume with a nonhomeomorphic hyperbolic 3-manifold. In addition, it is shown that two orientable finite volume hyperbolic 3-manifolds Mx and M2 containing incompressible thrice-punctured spheres Sx and S, respectively, can be cut open along S; and Sj and then glued together along copies of Sx and S2 to yield a 3-manifold which is hyperbolic with volume equal to the sum of the volumes of M1 and M2. Applications to link complements in S3 are included.