Thrice-punctured spheres in hyperbolic $3$-manifolds

Abstract:

The work of ${\text {W}}$. Thurston has stimulated much interest in the volumes of hyperbolic $3$-manifolds. In this paper, it is demonstrated that a $3$-manifold $M\prime$ 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 ${M_1}$ and ${M_2}$ containing incompressible thrice-punctured spheres ${S_1}$ and ${S_2}$, respectively, can be cut open along ${S_1}$ and ${S_2}$ and then glued together along copies of ${S_1}$ and ${S_2}$ to yield a $3$-manifold which is hyperbolic with volume equal to the sum of the volumes of ${M_1}$ and ${M_2}$. Applications to link complements in ${S^3}$ are included.