Let θ : π 1 (R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem.Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π 1 (R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to SL(2, C).
We investigate the relationship between an open simply-connected region Ω ⊂ S 2 and the boundary Y of the hyperbolic convex hull in H 3 of S 2 \ Ω. A counterexample is given to Thurston's conjecture that these spaces are related by a 2-quasiconformal homeomorphism which extends to the identity map on their common boundary, in the case when the homeomorphism is required to respect any group of Möbius transformations which preserves Ω. We show that the best possible universal lipschitz constant for the nearest point retraction r : Ω → Y is 2. We find explicit universal constants 0 < c 2 < c 1 , such that no pleating map which bends more than c 1 in some interval of unit length is an embedding, and such that any pleating map which bends less than c 2 in each interval of unit length is embedded. We show that every K-quasiconformal homeomorphism D 2 → D 2 is a (K, a(K))-quasi-isometry, where a(K) is an explicitly computed function. The multiplicative constant is best possible and the additive constant a(K) is best possible for some values of K.
We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded as a standard hyperbolic subspace. Given a complex number t, we obtain a corresponding object in hyperbolic 3-space by earthquaking along the lamination, parametrized by the real part of t, and then bending along the image lamination, parametrized by the complex part of t. In the literature, it is usually assumed that there is a quasifuchsian group that preserves the structure, but this paper is more general and makes no such assumption. Our deformation is holomorphic, as in the λ-lemma, which is a result that underlies the results in this paper. Our deformation is used to produce a new, more natural proof of Sullivan's theorem: that, under standard topological hypotheses, the boundary of the convex hull in hyperbolic 3-space of the complement of an open subset U of the 2-sphere is quasiconformally equivalent to U , and that, furthermore, the constant of quasiconformality is a universal constant. Our paper presents a precise statement of Sullivan's Theorem. We also generalize much of McMullen's Disk Theorem, describing certain aspects of the parameter space for certain parametrized spaces of 2-dimensional hyperbolic structures.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.