Flexible boundaries in deformations of hyperbolic 3-manifolds
For a complete finite-volume hyperbolic 3-manifold \(M\) with torus cusps Thurston considered deformations of the complete hyperbolic structure thus obtaining non-complete structures whose completions include almost all closed manifolds obtained by Dehn filling (surgery) on the cusps (the ``hyperbolic surgery theorem''). Now suppose that \(M\) has both cusps and compact totally geodesic boundary \(\partial M\). Then one may consider those deformations of the complete hyperbolic structure (that is complete on the double of \(M\) along \(\partial M)\) such that the boundary remains totally geodesic. By taking the induced hyperbolic structure on the boundary this defines a mapping \(B_M\) from the space of such deformations to the Teichmüller space of \(\partial M\). For dimension reasons, this mapping will not be injective in general. Neumann-Reid and the first author gave examples where it is constant. In contrast to this, in the present paper an infinite series of such manifolds \(M\) is constructed for which \(B_M\) is a local embedding at the complete structure. The manifolds are the \(n\)-fold cyclic branched coverings of hyperbolic orbifolds obtained from the complement of the Whitehead link by introducing a singular arc of index \(n\) whose endpoints are on the same component of the link. These orbifolds can be decomposed into 4 partially truncated ideal tetrahedra forming an octahedron. Then, using explicit parameters for the deformation space, it can be shown by a direct computation that the derivative of the map \(B_M\) has maximal rank.
scientific article; zbMATH DE number 1117470