Abstract
Let M be a cusped hyperbolic 3-manifold with non-empty geodesic boundary. A small Dehn filling deformation of M on the cusps can be performed so that the boundary is kept to be geodesic. Then assigning to each deformation a hyperbolic structure on the boundary, we get a map BM from the space of such deformations to the Teichmϋller space of dM. More precise argument for this fact will be given i n § l . Motivated by the conjectures posed in Cooper-Long [1] and Kapovich [5], Neumann and Reid [7] discovered many examples of M such that BM is a constant map. Fujii [3] also obtained another concrete family of small deformations for some M such that BM maps his family to a constant structure. These examples at first contrasted with our naive intuition that the deformation of hyperbolic structure affects everywhere. But the fact itself would not be too surprising once we realized that the dimension of the source can be bigger than that of the target and in that case BM can never be injective. A more pertinent problem to set up for the moment would be what the map BM looks like. In fact, we have known only a little about it so far. Under the circumstances above, it would be worth finding examples for which we can convert this rather difficult problem to something we can do by hand. In this paper, we will construct infinitely many one-cusped examples of M so that BM is a local embedding at the complete structure. The polyhedral construction will be discussed in §2. Then by using its polyhedral structure, we will compute the derivative of BM at the complete structure by hand in the later sections. The authors are indebted to Professor T. Soma for the argument in §1, and we would like to thank him.
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