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 volume, cusp shape, Heegaard genus, complex length of the shortest geodesic, length of the shortest return path, and Turaev-Viro invariants. Manifolds which share all these invariants are called geometrically similar, and were first studied by C. D. Hodgson, R. G. Meyerhoff and J. R. Weeks. The examples of geometrically similar manifolds they described are commensurable with each other. We show here that many elements in M_{g,k} admit non-commensurable geometrically similar Dehn fillings.
The notion of geometric isolation for cusps in a hyperbolic 3-manifold was introduced by W. D. Neumann and A. W. Reid and studied by D. Calegary, who provided explanations for all the previously known examples of isolation phenomena. We show here that the cusps of any manifold M_{g,k} are geometrically isolated from each other. Apparently, isolation of cusps in our examples arises for different reasons from those described by Calegari.
We also show that any element in M_{g,k} admits an infinite family of hyperbolic Dehn fillings inducing non-trivial deformations of the hyperbolic structure on the geodesic boundary.
Any one-cusped hyperbolic manifold M with an unknotting tunnel tau is obtained by Dehn filling a cusp of a two-cusped hyperbolic manifold. In the case where M is obtained by "generic" Dehn filling, we prove that tau is isotopic to a geodesic, and characterize whether tau is isotopic to an edge in the canonical decomposition of M. We also give explicit estimates (with additive error only) on the length of tau relative to a maximal cusp. These results give generic answers to three long-standing questions posed by Adams, Sakuma, and Weeks. We also construct an explicit sequence of one-tunnel knot…
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.
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.
A symplectic rational cuspidal curve with positive self-intersection number admits a concave neighborhood, and thus a corresponding contact manifold on the boundary. In this article, we study symplectic fillings of such contact manifolds, providing a complementary perspective to our earlier article on symplectic isotopy classes of rational cuspidal curves. We explore aspects of these symplectic fillings through Stein handlebodies and rational blow-downs. We give examples of such contact manifolds which are identifiable as links of normal surface singularities, other examples which admit no sym…
We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of M_{g,k} for a fixed k has growth type g^g. We completely describe the non-hyperbolic Dehn fillings of each M in M_{g,k}, showing that, on any cusp of any hyperbolic manifold obtained by partially filling M, the…
To a hyperbolic manifold one can associate a canonical projective structure and ask whether it can be deformed or not. In a cusped manifold, one can ask about the existence of deformations that are trivial on the boundary. We prove that if the canonical projective structure of a cusped manifold is infinitesimally projectively rigid relative to the boundary, then infinitely many Dehn fillings are projectively rigid. We analyze in more detail the figure eight knot and the Withehead link exteriors, for which we can give explicit infinite families of slopes with projectively rigid Dehn fillings.
Every cusped, finite-volume hyperbolic three-manifold has a canonical decomposition into ideal polyhedra. We study the canonical decomposition of the hyperbolic manifold obtained by filling some (but not all) of the cusps with solid tori: in a broad range of cases, generic in an appropriate sense, this decomposition can be predicted from that of the unfilled manifold. As an example, we treat all hyperbolic fillings on one cusp of the Whitehead link complement.
We classify all the non-hyperbolic Dehn fillings of the complement of the chain-link with 3 components, conjectured to be the smallest hyperbolic 3-manifold with 3 cusps. We deduce the classification of all non-hyperbolic Dehn fillings of infinitely many 1-cusped and 2-cusped hyperbolic manifolds, including most of those with smallest known volume. Among other consequences of this classification, we mention the following: - for every integer n we can prove that there are infinitely many hyperbolic knots in the 3-sphere having exceptional surgeries n, n+1, n+2, n+3, with n+1, n+2 giving small S…
It has been observed that most manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic $3$-manifolds are obtained by surgery on the minimally twisted 5-chain link. A full classification of the exceptional surgeries on the 5-chain link has recently been completed. In this article, we provide a complete classification of the sets of exceptional slopes and fillings for all cusped hyperbolic surgeries on the minimally twisted 5-chain link, thereby describing the sets of exceptional slopes and fillings for most hyperbolic manifolds of small complexity. The classification produces the…
Let M be a 1-cusped hyperbolic 3-manifold whose cusp shape is quadratic. We show that there exists c=c(M) such that the number of hyperbolic Dehn fillings of M with any given volume v is uniformly bounded by c.
This paper concerns with a rigidity of core geodesics in hyperbolic Dehn fillings. For instance, for an $n$-cusped hyperbolic $3$-manifold $M$ having non-symmetric cusp shapes, we show any Dehn filling of $M$ with sufficiently large coefficient is uniquely determined by the product of the holonomies of its core geodesics. We also explore various implications of the main results. An appendix by I. Agol provides an alternative geometric proof of one of the corollaries of our main arguments.
We show that there are at most finitely many one cusped orientable hyperbolic 3-manifolds which have more than eight non-hyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable.
The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3-manifolds, obtained by Dehn filling on the figure eight knot complement, that are uniquely determined by their volumes. This gives a sequence of distinct volumes x_i converging to the volume of the figure eight knot complement with N(x_i) = 1 for each i. We also give an infinite sequence of 1-cusped hyperbolic 3-manifolds, obtained by Dehn filling one cusp of the (-2,3,8)-pretzel li…
We prove an explicit, quantitative criterion that ensures the Heegaard surfaces in Dehn fillings behave "as expected." Given a cusped hyperbolic manifold X, and a Dehn filling whose meridian and longitude curves are longer than 2pi(2g-1), we show that every genus g Heegaard splitting of the filled manifold is isotopic to a splitting of the original manifold X. The analogous statement holds for fillings of multiple boundary tori. This gives an effective version of a theorem of Moriah-Rubinstein and Rieck-Sedgwick.
We classify the complete hyperbolic 3-manifolds admitting a maximal cusp of volume at most 2.62. We use this to show that the figure-8 knot complement is the unique 1-cusped hyperbolic 3-manifold with nine or more non-hyperbolic fillings; to show that the figure-8 knot complement and its sister are the unique hyperbolic 3-manifolds with minimal volume maximal cusps; and to extend results on determining low volume closed and cusped hyperbolic 3-manifolds.
This paper proves explicit bilipschitz bounds on the change in metric between the thick part of a cusped hyperbolic 3-manifold N and the thick part of any of its long Dehn fillings. Given a bilipschitz constant J > 1 and a thickness constant epsilon > 0, we quantify how long a Dehn filling suffices to guarantee a J-bilipschitz map on epsilon-thick parts. A similar theorem without quantitative control was previously proved by Brock and Bromberg, applying Hodgson and Kerckhoff's theory of cone deformations. We achieve quantitative control by bounding the analytic quantities that control the infi…
Let M be a hyperbolic n-manifold whose cusps have torus cross-sections. In arXiv:0901.0056, the authors constructed a variety of nonpositively and negatively curved spaces as "2π-fillings" of M by replacing the cusps of M with compact "partial cones" of their boundaries. These 2π-fillings are closed pseudomanifolds, and so have a fundamental class. We show that the simplicial volume of any such 2π-filling is positive, and bounded above by Vol(M)/v_n, where v_n is the volume of a regular ideal hyperbolic n-simplex. This result generalizes the fact that hyperbolic Dehn filling of a 3-manifold do…
Previous work of the authors with Bus Jaco determined a lower bound on the complexity of cusped hyperbolic 3-manifolds and showed that it is attained by the monodromy ideal triangulations of once-punctured torus bundles. This paper exhibits an infinite family of minimal ideal triangulations of Dehn fillings on the link $8^3_9$ that also attain this lower bound on complexity.
In order to obtain a closed orientable convex projective four-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic four-manifold through a continuous path of projective cone-manifolds.
An L-space is a rational homology 3-sphere with minimal Heegaard Floer homology. We give the first examples of hyperbolic L-spaces with no symmetries. In particular, unlike all previously known L-spaces, these manifolds are not double branched covers of links in S^3. We prove the existence of infinitely many such examples (in several distinct families) using a mix of hyperbolic geometry, Floer theory, and verified computer calculations. Of independent interest is our technique for using interval arithmetic to certify symmetry groups and non-existence of isometries of cusped hyperbolic 3-manifo…
We show that the set of cusp shapes of hyperbolic tunnel number one manifolds is dense in the Teichmuller space of the torus. A similar result holds for tunnel number n manifolds. As a consequence, for fixed n, there are infinitely many hyperbolic tunnel number n manifolds with at most one exceptional Dehn filling. This is in contrast to large volume Berge knots, which are tunnel number one manifolds, but with cusp shapes converging to a single point in Teichmuller space.
This paper is the second in a series whose goal is to understand the structure of low-volume complete orientable hyperbolic 3-manifolds. Using Mom technology, we prove that any one-cusped hyperbolic 3-manifold with volume <= 2.848 can be obtained by a Dehn filling on one of 21 cusped hyperbolic 3-manifolds. We also show how this result can be used to construct a complete list of all one-cusped hyperbolic three-manifolds with volume <= 2.848 and all closed hyperbolic three-manifolds with volume <= 0.943. In particular, the Weeks manifold is the unique smallest volume closed orientable hyperboli…
This paper gives a quantitative version of Thurston's hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of non-hyperbolic Dehn fillings on a cusped hyperbolic 3-manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic cone-manifold structures, using infinitesimal harmonic deformations and analysis of geometric limits.
We show that an immersed thrice-punctured sphere in a cusped orientable hyperbolic 3-manifold is either embedded or has a single clasp in a manifold obtained by hyperbolic Dehn filling on a cusp of the Whitehead link complement.
For a single cusped hyperbolic 3-manifold, Hodgson proved that there are only finitely many Dehn fillings of it whose trace fields have bounded degree. In this paper, we conjecture the same for manifolds with more cusps, and give the first positive results in this direction. For example, in the 2-cusped case, if a manifold has linearly independent cusp shapes, we show that the manifold has the desired property.To prove the results, we use the proof of the Bounded Height Conjecture in arithmetic geometry.
We use Heegaard splittings to give a criterion for a tunnel number one knot manifold to be non-fibered and to have large cyclic covers. We also show that such a knot manifold (satisfying the criterion) admits infinitely many virtually Haken Dehn fillings. Using a computer, we apply this criterion to the 2 generator, non-fibered knot manifolds in the cusped Snappea census. For each such manifold M, we compute a number c(M), such that, for any n>c(M), the n-fold cyclic cover of M is large.
We introduce and study some deformations of complete finite-volume hyperbolic four-manifolds that may be interpreted as four-dimensional analogues of Thurston's hyperbolic Dehn filling. We construct in particular an analytic path of complete, finite-volume cone four-manifolds $M_t$ that interpolates between two hyperbolic four-manifolds $M_0$ and $M_1$ with the same volume $\frac {8}3π^2$. The deformation looks like the familiar hyperbolic Dehn filling paths that occur in dimension three, where the cone angle of a core simple closed geodesic varies monotonically from $0$ to $2π$. Here, the sin…
We prove that any mapping torus of a pseudo-Anosov mapping class with bounded normalized Weil-Petersson translation length contains a finite set of transverse and level closed curves, and drilling out this set of curves results in one of a finite number of cusped hyperbolic 3-manifolds. The number of manifolds in the finite list depends only on the bound for normalized translation length. We also prove a complementary result that explains the necessity of removing level curves by producing new estimates for the Weil-Petersson translation length of compositions of pseudo-Anosov mapping classes …
We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1. We show that the manifolds in H(n) are hyperbolic (with geodesic boundary, and cusps corresponding to the loops), have least possible volume, and simplest boundary loops. Mirroring the elements of H(n) in their geodesic boundary we obtain a class D(n) of cusped hyperbolic manifolds, previously considered by D. Thurston and the first named author. We show tha…
This paper gives the first explicit, two-sided estimates on the cusp area of once-punctured torus bundles, 4-punctured sphere bundles, and 2-bridge link complements. The input for these estimates is purely combinatorial data coming from the Farey tesselation of the hyperbolic plane. The bounds on cusp area lead to explicit bounds on the volume of Dehn fillings of these manifolds, for example sharp bounds on volumes of hyperbolic closed 3-braids in terms of the Schreier normal form of the associated braid word. Finally, these results are applied to derive relations between the Jones polynomial …
We give new bounds for the distance between two exceptional filling slopes for a 1-cusped hyperbolic 3-manifold in several different situations. The distance between a reducible slope and a slope that produces a manifold with finite fundamental group is at most 2. The distance between a reducible slope and one that produces a very small manifold is also at most 2. The distance between a reducible slope and one which produces a manifold with a pi_1 injective torus is at most 4. The methods used involve both characteristic submanifold theory and the theory of the PSL(2,C) character variety.
We give new examples of hyperbolic and relatively hyperbolic groups of cohomological dimension $d$ for all $d\geq 4$. These examples result from applying CAT$(0)$/CAT$(-1)$ filling constructions (based on singular doubly warped products) to finite volume hyperbolic manifolds with toral cusps. The groups obtained have a number of interesting properties, which are established by analyzing their boundaries at infinity by a kind of Morse-theoretic technique, related to but distinct from ordinary and combinatorial Morse theory.
Let $F$ be a closed essential surface in a hyperbolic 3-manifold $M$ with a toroidal cusp $N$. The depth of $F$ in $N$ is the maximal distance from points of $F$ in $N$ to the boundary of $N$. It will be shown that if $F$ is an essential pleated surface which is not coannular to the boundary torus of $N$ then the depth of $F$ in $N$ is bounded above by a constant depending only on the genus of $F$. The result is used to show that an immersed closed essential surface in $M$ which is not coannular to the torus boundary components of $M$ will remain essential in the Dehn filling manifold $M(γ)$ a…
Extending earlier work of Tian, we show that if a manifold admits a metric that is almost hyperbolic in a suitable sense, then there exists an Einstein metric that is close to the given metric in the $C^{2,α}$-topology. In dimension $3$ the original manifold only needs to have finite volume, and the volume can be arbitrarily large. Applications include a new proof of the hyperbolization of $3$-manifolds of large Hempel distance yielding some new geometric control on the hyperbolic metric, and an analytic proof of Dehn filling and drilling that allows the filling and drilling of arbitrary many …
We give effective bilipschitz bounds on the change in metric between thick parts of a cusped hyperbolic 3-manifold and its long Dehn fillings. In the thin parts of the manifold, we give effective bounds on the change in complex length of a short closed geodesic. These results quantify the filling theorem of Brock and Bromberg, and extend previous results of the authors from finite volume hyperbolic 3-manifolds to any tame hyperbolic 3-manifold. To prove the main results, we assemble tools from Kleinian group theory into a template for transferring theorems about finite-volume manifolds into th…
We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one, filling which one gets a wealth of hyperbolic 3-manifolds having 4 or fewer (including 0) cusps. In view of Thurston's hyperbolic Dehn filling theorem it is then natural to face the problem of classifying all the exceptional fillings on M5, namely those yielding non-hyperbolic 3-manifolds. Here we completely solve this problem, also showing that, thanks to the s…
For $d=4, 5, 6$, we exhibit the first examples of complete finite volume hyperbolic $d$-manifolds $M$ with cusps such that infinitely many $d$-orbifolds $M_{m}$ obtained from $M$ by generalized Dehn filling admit properly convex real projective structures. The orbifold fundamental groups of $M_m$ are Gromov-hyperbolic relative to a collection of subgroups virtually isomorphic to $\mathbb{Z}^{d-2}$, hence the images of the developing maps of the projective structures on $M_m$ are new examples of divisible properly convex domains of the projective $d$-space which are not strictly convex, in cont…
Neumann and Reid conjecture that there are exactly three knot complements which admit hidden symmetries. This paper establishes several results that provide evidence for the conjecture. Our main technical tools provide obstructions to having infinitely many fillings of a cusped manifold produce knot complements admitting hidden symmetries. Applying these tools, we show for any two-bridge link complement, at most finitely many fillings of one cusp can be covered by knot complements admitting hidden symmetries. We also show that the figure-eight knot complement is the unique knot complement with…
We show that, for any given 3-manifold M, there are at most finitely many hyperbolic knots K in the 3-sphere and fractions p/q (with q > 22), such that M is obtained by p/q surgery along K. This is a corollary of the following result. If M is obtained by Dehn filling the cusps of a hyperbolic 3-manifold X, where each filling slope has length more than 2 π+ ε, then, for any given M and ε> 0, there are only finitely many possibilities for X and for the filling slopes. In this paper, we also investigate the length of boundary slopes, and sequences of negatively curved metrics on a given 3-manifol…
We study the IR phases of 3D class R theories associated with closed non-hyperbolic 3-manifolds. Non-hyperbolic 3-manifolds can be obtained by performing Dehn fillings on 1-cusped hyperbolic 3-manifolds along exceptional slopes. In 3D-3D correspondence, the `exceptional' Dehn filling corresponds to the gauging of an $SU(2)$ flavor symmetry in a superconformal field theory associated with a 1-cusped 3-manifold with `small' Chern-Simons levels. With several explicit examples, we analyze various interesting non-perturbative IR phenomena (such as spontaneous SUSY breaking, generation of mass gap a…
This paper concerns the set $\hat{\mathcal{M}}$ of pseudo-Anosovs which occur as monodromies of fibrations on manifolds obtained from the magic 3-manifold $N$ by Dehn filling three cusps with a mild restriction. We prove that for each $g$ (resp. $g \not\equiv 0 \pmod{6}$), the minimum among dilatations of elements (resp. elements with orientable invariant foliations) of $\hat{\mathcal{M}}$ defined on a closed surface $\varSigma_g$ of genus $g$ is achieved by the monodromy of some $\varSigma_g$-bundle over the circle obtained from $N(\tfrac{3}{-2})$ or $N(\tfrac{1}{-2})$ by Dehn filling two cus…
We show that every hyperbolic link complement contains closed quasi-Fuchsian surfaces. As a consequence, we obtain the result that on a hyperbolic link complement, if we remove from each cusp of the manifold a certain finite set of slopes, then all remaining Dehn fillings on the link complement yield manifolds with closed immersed incompressible surfaces.
