A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation preserving the cusp. The bounds are in terms of the change in structure in a neighborhood of the singular locus alone.
We then apply this result to provide information on the cusp shape of many hyperbolic knots, given only a diagram of the knot. Specifically, we show there is a universal constant C such that if a knot admits a prime, twist reduced diagram with at least C crossings per twist region, then the length of the second shortest curve on the cusp torus is bounded. The bound is linear in the number of twist regions of the diagram.
Let f be a 1-variable complex polynomial such that f has a singularity at the origin. In the present paper, we show that there exists a deformation of f which has only fold singularities and cusps as singularities of a real polynomial map from the plane to the plane. We then calculate the number of cusps of a deformation in a sufficiently small neighborhood of the origin.
In this paper, we study deformations of Brieskorn polynomials of two variables obtained by adding linear terms consisting of the conjugates of complex variables and prove that the deformed polynomial maps have only indefinite fold and cusp singularities in general. We then estimate the number of cusps appearing in such a deformation. As a corollary, we show that a deformation of a complex Morse singularity with real linear terms has only indefinite folds and cusps in general and the number of cusps is 3.
This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence $(M_{i}%, p_{i}) $ of pointed hyperbolic cone-manifolds with topological type $(M,Σ) $, where $M$ is a closed, orientable and irreducible 3-manifold and $Σ$ an embedded link in $M$. If the sequence $M_{i}$ collapses and assuming that the lengths of the singularity remain uniformly bounded, we prove that $M$ is either a Seifert fibered or a $Sol$ manifold. We apply this result to a question stated…
We give an expository account of our proof that each cusp-free hyperbolic 3-manifold M with finitely generated fundamental group and incompressible ends is an algebraic limit of geometrically finite hyperbolic 3-manifolds.
Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-manifold is decreasing and that summed with the cone angle squared is increasing as we deform the cone-angles. We confirm this near 0 cone-angles for an infinite family of hyperbolic cone-manifolds obtained by Dehn surgeries along the Whitehead link complements. The basic method is…
Two single parameter families of polyhedra $P(ψ)$ are constructed in three dimensional spaces of constant curvature $C(ψ)$. Identification of the faces of the polyhedra via isometries results in cone manifolds $M(ψ)$ which are topologically $S^1\timesS^2$, $S^3$ or singular $S^2$. The singular set of $M(ψ)$ can have self intersections for some values of $ψ$ and can also be the Whitehead link or form other configurations. Curvature varies continuously with $ψ$. At $ψ=0$ spontaneous surgery occurs and the topological type of $M(ψ)$ changes. This phenomenon is described.
We develop the deformation theory of hyperbolic cone-3-manifolds with cone-angles less than $2π$, i.e. contained in the interval $(0,2π)$. In the present paper we focus on deformations keeping the topological type of the cone-manifold fixed. We prove local rigidity for such structures. This gives a positive answer to a question of A. Casson.
This work is devoted to the study of deformations of hyperbolic cone structures under the assumption that the lengths of the singularity remain uniformly bounded over the deformation. Given a sequence (M_{i},p_{i}) of pointed hyperbolic cone-manifolds with topological type (M,Σ), where M is a closed, orientable and irreducible 3-manifold and Σ an embedded link in M. Assuming that the lengths of the singularity remain uniformly bounded, we prove that either the sequence M_{i} collapses and M is Seifert fibered or a Sol manifold, or the sequence M_{i} does not collapse and in this case a subsequ…
Let $ M$ be a cusped hyperbolic $ 3$-manifold, e.g. a knot complement. Thurston showed that the space of deformations of its fundamental group in $ \mathrm {PGL}(2,\mathbf {C})$ (up to conjugation) is of complex dimension the number $ ν$ of cusps near the hyperbolic representation. It seems natural to ask whether some representations remain discrete after deformation. The answer is generically not. A simple reason for it lies inside the cusps: the degeneracy of the peripheral representation (i.e. representations of fundamental groups of the $ ν$ peripheral tori). They indeed generically become…
We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles $\le π$), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotient of the one-holed torus by its elliptic involution, and the closed genus two surface by its hyper-elliptic involution, we obtain generalizations of the Weierstrass identities for the one-holed torus, and identities for the genus two surface, also obtained by McShane using differ…
We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a…
We show that any compact orientable hyperbolic 3-cone-manifold with cone angle at most πcan be continuously deformed to a complete hyperbolic manifold homeomorphic to the complement of the singularity. This together with the local rigidity by Hodgson and Kerckhoff implies the global rigidity for compact orientable hyperbolic 3-cone-manifolds under the same angle assumption.
We prove 3-dimensional hyperbolic cone-manifolds are geometrically inflexible: a cone-deformation of a hyperbolic cone-manifold determines a bi-Lipschitz diffeomorphism between initial and terminal manifolds in the deformation in the complement of a standard tubular neighborhood of the cone-locus whose pointwise bi-Lipschitz constant decays exponentially in the distance from the cone-singularity. Estimates at points in the thin part are controlled by similar estimates on the complex lengths of short curves.
In two former papers, the authors independently proved that the space of hyperbolic cone-3-manifolds with cone angles less than 2π and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, i.e. when the singular vertices can be split. We show that the different possible splittings correspond to specific pair-of-pants decompositions of the smooth parts of the links of the singular vertices, and that under suitable assumptions the corresponding subspace of deformations i…
The deformation theory of hyperbolic and Euclidean cone-manifolds with all cone angles less then 2π plays an important role in many problems in low dimensional topology and in the geometrization of 3-manifolds. Furthermore, various old conjectures dating back to Stoker about the moduli of convex hyperbolic and Euclidean polyhedra can be reduced to the study of deformations of cone-manifolds by doubling a polyhedron across its faces. This deformation theory has been understood by Hodgson and Kerckhoff when the singular set has no vertices, and by Weiß when the cone angles are less than π. We pr…
Neumann and Reid described in their paper "Rigidity of cusps in deformations of hyperbolic 3-orbifolds" (Math Ann. 295 (1993) no. 2, 223--237) a 2-cusped hyperbolic 3-orbifold in which the cusps are geometrically isolated. Based on numerical evidence provided by Jeff Weeks' snappea program, they conjectured that the cusps are strongly geometrically isolated, a fact which we establish here. We also give a parameterization of the Dehn surgery space of this orbifold which has amusing properties.
Napoleon's theorem in elementary geometry describes how certain linear operations on plane polygons of arbitrary shape always produce regular polygons. More generally, certain triangulations of a polygon that tiles R^2 admit deformations which keep fixed the symmetry group of the tiling. This gives rise to isolation phenomena in cusped hyperbolic 3-manifolds, where hyperbolic Dehn surgeries on some collection of cusps leaves the geometric structure at some other collection of cusps unchanged.
We prove that non-compact finite volume hyperbolic 3-manifolds that satisfy a mild cohomological condition (infinitesimal rigidity) admit a family of properly convex deformations of their complete hyperbolic structure where the ends become generalized cusps of type 1 or type 2. We also discuss methods for controlling which types of cusp occur. Using these methods we produce the first known example of a 1-cusped hyperbolic 3-manifold that admits a convex projective structure with a type 2 cusp. We also use these techniques to produce new 1-cusped manifolds that admit a convex projective structu…
This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory. A central idea is that local rigidity results (for deformations fixing cone angles) can be turned into effective control on the deformations that do exist. This leads to precise analytic and geometric versions of the idea that hyperbolic structures with short geodesics are close to hyperbolic structures with cusps. The paper also outlines a new harmonic defo…
We prove that every finite-volume hyperbolic 3-manifold M with p > 0 cusps admits a canonical, complete, piecewise Euclidean CAT(0) metric, with a canonical projection to a CAT(0) spine K. Moreover, (a) the universal cover of M endowed with the CAT(0) metric is a union of Euclidean half-spaces, glued together by identifying Euclidean polygons in their bounding planes by pairwise isometry (b)each cusp of M in the CAT(0) metric is a non-singular metric product of a (Euclidean) cusp torus and a half-line (c) all metric singularities are concentrated on the 1-skeleton of K, with cone angles a mult…
We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive reals with a 3-dimensional tower formed by deformations of lattices in the ring of integers of K, and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples.
In this paper we deepen the analysis of certain classes M_{g,k} of hyperbolic 3-manifolds that were introduced in a previous work by B. Martelli, C. Petronio and the author. Each element of M_{g,k} is an oriented complete finite-volume hyperbolic 3-manifold with compact connected geodesic boundary of genus g and k cusps. We study small deformations of the complete hyperbolic structure of manifolds in M_{g,k} via a close analysis of their geodesic triangulations. We prove that several elements in M_{g,k} admit non-homeomorphic hyperbolic Dehn fillings sharing the same volume, homology, cusp vol…
We study the geometry of hyperbolic cone surfaces, possibly with cusps or geodesic boundaries. We prove that any hyperbolic cone structure on a surface of non-exceptional type is determined up to isotopy by the geodesic lengths of a finite specific homotopy classes of non-peripheral simple closed curves. As an application, we show that the Thurston asymmetric metric is well-defined on the Teichmüller space of hyperbolic cone surfaces with fixed cone angles and boundary lengths. We compare such a Teichmüller space with the Teichmüller space of complete hyperbolic surfaces with punctures, by sho…
