Complete Hyperbolic Structure Research Articles

Limits of sequences of pseudo-Anosov maps and of hyperbolic 3–manifolds

There are two objects naturally associated with a braid $\beta\in B_n$ of pseudo-Anosov type: a (relative) pseudo-Anosov homeomorphism $\varphi_\beta\colon S^2\to S^2$; and the finite volume complete hyperbolic structure on the 3-manifold $M_\beta$ obtained by excising the braid closure of $\beta$, together with its braid axis, from $S^3$. We show the disconnect between these objects, by exhibiting a family of braids $\{\beta_q:q\in{\mathbb{Q}}\cap(0,1/3]\}$ with the properties that: on the one hand, there is a fixed homeomorphism $\varphi_0\colon S^2\to S^2$ to which the (suitably normalized) homeomorphisms $\varphi_{\beta_{q}}$ converge as $q\to 0$; while on the other hand, there are infinitely many distinct hyperbolic 3-manifolds which arise as geometric limits of the form $\lim_{k\to\infty} M_{\beta_{q_k}}$, for sequences $q_k\to 0$.

Constructing convex projective 3‐manifolds with generalized cusps

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 structure with a type 1 cusp.

Automorphisms of Two-Generator Free Groups and Spaces of Isometric Actions on the Hyperbolic Plane

The automorphisms of a two-generator free group acting on the space of orientation-preserving isometric actions of on hyperbolic 3-space defines a dynamical system. Those actions which preserve a hyperbolic plane but not an orientation on that plane is an invariant subsystem, which reduces to an action on R^3 by polynomial automorphisms preserving the cubic polynomial and an area form on the level surfaces. We describe the dynamical decomposition of this action. The domain of discontinuity of this action corresponds to geometric structures: either complete hyperbolic structures on the 2-holed cross-surface (projective plane) with cusps and funnels, or complete hyperbolic structures on a one-holed Klein bottle, or hyperbolic structures on a Klein bottle with one conical singularity. The action is ergodic on the complement of the orbit of the Fricke space of the 2-holed cross-surface, and we show that the orbit of the generalized Fricke space of the one-holed Klein bottle is open and dense.

Real structures on marked Schottky space

Schottky groups are exactly those Kleinian groups providing the regular lowest planar uniformizations of closed Riemann surfaces and also the ones providing to the interior of a handlebody of a complete hyperbolic structure with injectivity radius bounded away from zero. The space parametrizing quasiconformal deformations of Schottky groups of a fixed rank $g \geq 1$ is the marked Schottky space ${\mathcal M}{\mathcal S}_{g}$; this being a complex manifold of dimension $3(g-1)$ for $g \geq 2$ and being isomorphic to the punctured unit disc for $g=1$. In this paper we provide a complete description of the real structures of ${\mathcal M}{\mathcal S}_{g}$, up to holomorphic automorphisms, together their real part.

An application of non-positively curved cubings of alternating links

By using non-positively curved cubings of prime alternating link exteriors, we prove that certain ideal triangulations of their complements, derived from reduced alternating diagrams, are non-degenerate, in the sense that none of the edges is homotopic relative its endpoints to a peripheral arc. This guarantees that the hyperbolicity equations for those triangulations for hyperbolic alternating links have solutions corresponding to the complete hyperbolic structures. Since the ideal triangulations considered in this paper are often used in the study of the volume conjecture, this result has a potential application to the volume conjecture.

Group actions, Teichmüller spaces and cobordisms

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism M whose interior has a complete hyperbolic structure depend on properties of the variety of discrete representations of the fundamental group of its 3-dimensional boundary ∂M. In addition to the standard conformal ergodic action of a uniformhyperbolic lattice on the round sphere S n−1 and its quasiconformal deformations in S n , we present several constructions of unusual actions of such lattices on everywhere wild spheres (boundaries of quasisymmetric embeddings of the closed n-ball into S n ), on non-trivial (n − 1)-knots in S n+1, as well as actions defining non-trivial compact cobordisms with complete hyperbolic structures in its interiors. We show that such unusual actions always correspond to discrete representations of a given hyperbolic lattice from “non-standard” components of its varieties of representations (faithful or with large kernels of defining homomorphisms).

Verified Computations for Hyperbolic 3-Manifolds

For a given cusped 3-manifold M admitting an ideal triangulation, we describe a method to rigorously prove that either M or a filling of M admits a complete hyperbolic structure via verified computer calculations. Central to our method is an implementation of interval arithmetic and Krawczyk’s test. These techniques represent an improvement over existing algorithms as they are faster while accounting for error accumulation in a more direct and user-friendly way.

The Colored Jones Polynomial, the Chern–Simons Invariant, and the Reidemeister Torsion of a Twice–Iterated Torus Knot

A generalization of the volume conjecture relates the asymptotic behavior of the colored Jones polynomial of a knot to the Chern–Simons invariant and the Reidemeister torsion of the knot complement associated with a representation of the fundamental group to the special linear group of degree two over complex numbers. If the knot is hyperbolic, the representation can be regarded as a deformation of the holonomy representation that determines the complete hyperbolic structure. In this article, we study a similar phenomenon when the knot is a twice-iterated torus knot. In this case, the asymptotic expansion of the colored Jones polynomial splits into sums, and each summand is related to the Chern–Simons invariant and the Reidemeister torsion associated with a representation.

Deformations of noncompact projective manifolds

In this paper, we demonstrate that the complete hyperbolic structure of various two-bridge knots and links cannot be deformed to an inequivalent strictly convex projective structure. We also prove a complementary result showing that under certain rigidity hypotheses, branched covers of amphicheiral knots admit non-trivial, strictly convex deformations near their complete hyperbolic structure.

A variation of McShane’s identity for 2–bridge links

We give a variation of McShane's identity, which describes the cusp shape of a hyperbolic 2-bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL(2,\mathbb{C})$-characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of 2-bridge link complements.

The quantum content of the gluing equations

The gluing equations of a cusped hyperbolic 3‐manifold M are a system of polynomial equations in the shapes of an ideal triangulation T of M that describe the complete hyperbolic structure of M and its deformations. Given a Neumann‐Zagier datum (comprising the shapes together with the gluing equations in a particular canonical form) we define a formal power series with coefficients in the invariant trace field of M that should (a) agree with the asymptotic expansion of the Kashaev invariant to all orders, and (b) contain the nonabelian Reidemeister‐Ray‐Singer torsion of M as its first subleading “1‐loop” term. As a case study, we prove topological invariance of the 1‐loop part of the constructed series and extend it into a formal power series of rational functions on the PSL.2;C/ character variety of M . We provide a computer implementation of the first three terms of the series using the standard SnapPy toolbox and check numerically the agreement of our torsion with the Reidemeister‐Ray‐Singer for all 59924 hyperbolic knots with at most 14 crossings. Finally, we explain how the definition of our series follows from the quantization of 3‐dimensional hyperbolic geometry, using principles of topological quantum field theory. Our results have a straightforward extension to any 3‐manifold M with torus boundary components (not necessarily hyperbolic) that admits a regular ideal triangulation with respect to some PSL.2;C/ representation.

Proving a manifold to be hyperbolic once it has been approximated to be so

The computer program SnapPea can approximate whether or not a three manifold whose boundary consists of tori has a complete hyperbolic structure, but it can not prove conclusively that this is so. This article provides a method for proving that such a manifold has a complete hyperbolic structure based on the approximations of SNAP, a program that includes the functionality of SnapPea plus other features. The approximation is done by triangulating the manifold, identifying consistency and completeness equations with respect to this triangulation, and then trying to solve the system of equations using Newton's Method. This produces an approximate, not actual solution. The method developed here uses Kantorovich's theorem to prove that an actual solution exists, thereby assuring that the manifold has a complete hyperbolic structure. Using this, we can definitively prove that every manifold in the SnapPea cusped census has a complete hyperbolic structure.

Generalized volume conjecture and the [formula omitted]-polynomials: The Neumann–Zagier potential function as a classical limit of the partition function

We introduce and study the partition function Z γ ( M ) for the cusped hyperbolic 3-manifold M . We construct formally this partition function based on an oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in his studies of the modular double of the quantum group. Following Thurston and Neumann–Zagier, we deform a complete hyperbolic structure of M , and we define the partition function Z γ ( M u ) correspondingly. This function is shown to give the Neumann–Zagier potential function in the classical limit γ → 0 , and the A -polynomial can be derived from the potential function. We explain our construction by taking examples of 3-manifolds such as complements of hyperbolic knots and a punctured torus bundle over the circle.

The Cannon–Thurston map for punctured-surface groups

Let Γ be the fundamental group of a compact surface group with non-empty boundary. We suppose that Γ admits a properly discontinuous strictly type preserving action on hyperbolic 3-space such that there is a positive lower bound on the translation lengths of loxodromic elements. We describe the Cannon–Thurston map in this case. In particular, we show that there is a continuous equivariant map of the circle to the boundary of hyperbolic 3-space, where the action on the circle is obtained by taking any finite-area complete hyperbolic structure on the surface, and lifting to the boundary of hyperbolic 2-space. We deduce that the limit set is locally connected, hence a dentrite in the singly degenerate case. Moreover, we show that the Cannon–Thurston map can be described topologically as the quotient of the circle by the equivalence relations arising from the ends of the quotient 3-manifold. For closed surface bundles over the circle, this was obtained by Cannon and Thurston. Some generalisations and variations have been obtained by Minsky, Mitra, Alperin, Dicks, Porti, McMullen and Cannon. We deduce that a finitely generated kleinian group with a positive lower bound on the translation lengths of loxodromics has a locally connected limit set assuming it is connected.

Totally geodesic boundaries of knot complements

Given a compact orientable 3 3 –manifold M M whose boundary is a hyperbolic surface and a simple closed curve C C in its boundary, every knot in M M is homotopic to one whose complement admits a complete hyperbolic structure with totally geodesic boundary in which the geodesic representative of C C is as small as you like.

Peripheral polynomials of hyperbolic knots

If K is a hyperbolic knot in S 3 , an algebraic component of its character variety containing one holonomy of the complete hyperbolic structure of finite volume of S 3 ∖ K is an algebraic curve K . The traces of the peripheral elements of K define polynomial functions in K , which are related in pairs by polynomials ( peripheral polynomials). These are determined by just two adjacent peripheral polynomials. The curves defined by the peripheral polynomials are all birationally equivalent to K , with only one possible exception. The canonical peripheral polynomial relating the trace of the meridian with the trace of the canonical longitude of K, is a factor of the A-polynomial.

Degeneration of hyperbolic structures on the figure-eight knot complement and points of finite order on an elliptic curve

In this paper, we consider a hyperbolic structure on a manifold to be a Riemannian metric of constant sectional curvature −1 which is not necessarily complete. A hyperbolic 3-manifold is an orientable 3-manifold with a hyperbolic structure. It is well known that the complement of the figure-eight knot K in the 3-dimensional sphere S admits a complete, finite volume hyperbolic structure σ∞. By Mostow-Prasad rigidity, such a hyperbolic structure on S − K is unique. Incomplete hyperbolic structures are also of interest. Indeed, Thurston [8] analyzed the flexibility of hyperbolic structures on S − K by allowing incomplete hyperbolic structures. In fact, he showed that there are deformations of σ∞ on S3−K that are not complete hyperbolic structures. Such deformations are holomorphically parametrized by points in an open subset U of a complex affine plane curve C. The curve C and this subset U are given in (1) and (2) below. When the parameter of deformation becomes close to the boundary ∂U or enters C−U , degeneration of hyperbolic structures on S3−K occurs. There are cases in which such degeneration results in closed 3-manifolds obtained by Dehn fillings of S −K with other types of geometric structure. In fact, twenty such cases have been identified. In each case, this resultant closed 3-manifold is one of the following types: a Sol-manifold, a PSL2(R)-manifold, a Haken manifold which is decomposed into a PSL2(R)-manifold and a Euclidean manifold along an embedded torus, or a Euclidean orbifold. The genus of the curve C is one, because C is of degree four and has two ordinary double points. In fact, we give an explicit form of a birational map from C to a non-singular plane cubic curve in Weierstrass form E, which is given in (4) below. The curve E is an example of what is called an ‘elliptic curve defined over Q’. It is well known that any elliptic curve is an abelian group under an addition law. In this paper, we see that there is a concrete

CHARACTER VARIETIES AND PERIPHERAL POLYNOMIALS OF A CLASS OF KNOTS

For a hyperbolic knot, the excellent curve is the union of components of the character variety containing the characters of the complete hyperbolic structures on the complement of the knot. The peripheral polynomials define the excellent curve in terms of traces of bases of the boundary torus around the knot. In this paper we introduce the concept of net presentation of a knot with 2n strings which generalises the well known plat presentation of a knot with 2n strings. Nets with 4 strings are the correct setting to apply quaternion methods to compute the excellent curve. This is done here in a convenient way for application to the study of invariants of cone-manifold structures.

Classes of separable two-generator free subgroups of 3-manifold groups

In this paper, we consider subgroup separability of free products with amalgamation of Kleinian groups of finite covolume. Specifically, we prove the following. Let M1 and M2 be compact orientable 3-manifolds with non-empty boundary whose interiors admit complete hyperbolic structures of finite volume. Fix boundary components T1∈∂M1 and T2∈∂M2, and let f:T1→T2 be a homeomorphism. Let M be the manifold obtained by identifying M1 and M2 along the fixed boundary components via f. If α and β are loxodromic elements of π1(M1) and π1(M2), respectively, then 〈α,β〉 is a separable subgroup of π1(M).

Crossing change and exceptional Dehn surgery

Thurston’s hyperbolic Dehn surgery theorem [11], [12] asserts that if a knot in the 3-sphere 3 is hyperbolic (i.e., 3 − admits a complete hyperbolic structure of finite volume), then all but finitely many Dehn surgeries on yield hyperbolic 3-manifolds. By an exceptional surgery on a hyperbolic knot we mean a nontrivial Dehn surgery producing a non-hyperbolic manifold. Refer to [3], [6] for a survey on Dehn surgery on knots. We empirically know that ‘most’ knots are hyperbolic and ‘most’ hyperbolic knots have no exceptional surgeries. In this paper, we demonstrate the abundance of hyperbolic knots with no exceptional surgeries by showing that every knot is ‘close’ to infinitely many such hyperbolic knots in terms of crossing change. We regard that two knots are the same if they are isotopic in 3. For a knot in 3, let ( ) be the set of knots each of which is obtained by changing at most crossings in a diagram of .