All Questions
73 questions
8
votes
1
answer
223
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
1
vote
1
answer
91
views
When is a 2-bridge knot hyperbolic?
It is known that 2-bridge knots in $S^3$ can be classified by the Schubert form. My question is: which 2-bridge knots are hyperbolic? (Do we have a complete classification for hyperbolicity in 2-...
- 605
10
votes
0
answers
139
views
Space of thick ending laminations
Let $\Sigma$ be a compact closed connected oriented surface of genus $g>1$. Klarreich proved that the space of ending laminations $\mathcal{EL}(\Sigma)$ is the ideal boundary of the curve complex $...
- 68.9k
8
votes
1
answer
352
views
Can I endow the following 3-manifold with a hyperbolic metric?
Consider the following three-dimensional topology. Start with $S^3$ and drill out four unlinked tori as shown in the picture. Then, fill in the gaps with the same tori but with their longitudes and ...
- 355
3
votes
1
answer
202
views
Guts of 3-manifolds for sutured manifolds and pared manifolds
I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me.
...
- 502
5
votes
1
answer
182
views
Volume of the Weeks manifold and of the 5.2 knot complement
Some computations show that the Weeks manifold and the 5.2 knot complement have the same trace field (which is $\mathbb{Q}[x]/(x^3-x+1)$) and the (hyperbolic) volume of the second is 3 times the ...
- 1,181
2
votes
1
answer
350
views
Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries
By Bers' simultaneous uniformization theorem, if $\Gamma$ is a Fuchsian group, then $\operatorname{QC}(\Gamma)\cong \mathcal{T}(S)\times\mathcal{T}(\overline{S})$ where $S = \Bbb H^2/\Gamma$. In ...
3
votes
0
answers
72
views
Discreteness of volumes of boundary-parabolic representations
Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
- 2,533
5
votes
3
answers
245
views
Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$
Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
- 2,533
8
votes
1
answer
660
views
On trivial mapping class group of 3-manifolds
What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
- 3,751
3
votes
1
answer
239
views
Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?
It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed ...
- 33
10
votes
3
answers
493
views
Small examples of exceptional hyperbolic Dehn Filling of hyperbolic manifolds
For experimental purposes, I would like to have a small (i.e. triangulated with few tetrahedra) example of a manifold $M$ with the following properties:
$M$ is a hyperbolic manifold with finite ...
- 829
8
votes
0
answers
432
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
- 111
4
votes
0
answers
172
views
Survey or good reference of taut foliations
I am interested in the topology of foliations.
In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows.
I guess that
A. Candel and L. Conlon, Foliations I (...
- 73
4
votes
1
answer
177
views
Complex length of geodesic added in hyperbolic Dehn surgery
Suppose $M$ is a cusped finite-volume hyperbolic $3$-manifold, say with a single cusp for simplicity. Following [NZ, Section 4] we can parametrize deformations of the hyperbolic structure with a ...
- 2,533
3
votes
1
answer
363
views
Volume of hyperbolic 3-manifolds with toroidal boundary
A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.
This statement is from 3-Manifold Groups, page 18 (...
- 502
17
votes
3
answers
2k
views
Examples of the Thurston geometries with transitive Lie group action
Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries:
(1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup
(2) Euclidean: 3 torus $\...
- 4,081
3
votes
2
answers
344
views
Hyperbolic volume of hyperbolic knots
Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...
- 223
4
votes
1
answer
195
views
Ideal triangulation of hyperbolic 3-manifold with generic mapping class group
I am from physics background so I apologize in advance if my question is trivial.
Kojima proves for every finite group $G$, there is a hyperbolic 3-manifold such that its mapping class group equals $G$...
- 63
4
votes
2
answers
306
views
Quadratic cusp shape
Which hyperbolic $3$-manifolds are known to have quadratic cusp shape?
Explanations: Cusps of hyperbolic $3$-manifolds are products torus x interval. They lift to horoballs in hyperbolic $3$-space, ...
- 10.4k
3
votes
1
answer
176
views
Special kind of 3-manifolds
Is there an open connected orientable 3-manifold $M$ with the following properties:
$M$ admits a complete hyperbolic metric with finite hyperbolic volume.
$H_{i}(M,\mathbb{Z})=0$ for any $i>0$.
- 223
1
vote
2
answers
335
views
The moduli space of finite volume hyperbolic 3-manifolds?
By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$.
I will call
$$\mathcal{M}...
- 223
1
vote
0
answers
397
views
References on Hyperbolic Geometry and Teichmuller Theory
I am asking a soft question here.
I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
user318929
3
votes
0
answers
244
views
Hyperbolic metrics and the general Ahlfors-Bers theorem
Let $M$ be an oriented smooth compact 3-manifold with non-empty boundary and hyperbolizable interior such that all boundary components have genus greater than $1$. Denote $N:={\rm int}(M)$ and
$$HM_{...
- 353
5
votes
1
answer
297
views
Hyperbolicity of twist knots
In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the ...
user160180
3
votes
1
answer
202
views
Maximally symmetric hyperbolic 3-manifolds with finite volume
In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...
- 33
6
votes
1
answer
466
views
Hyperbolic manifolds with infinite cyclic fundamental group
It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...
- 2,533
7
votes
1
answer
204
views
A coincidence between the Lambert cube, Lobell polyhedron, and hyperbolic 3-manifolds?
I. Lambert cube $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$
In this paper (p.8), we find the volume $V$ of the hyperbolic Lambert cube for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as
$...
- 12.6k
9
votes
2
answers
886
views
Hyperbolic $3$-manifold groups that embed in compact Lie groups
Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...
- 29.1k
5
votes
1
answer
245
views
Intersection of $\pi_1$-injective surfaces
Let $M$ be a closed non-Haken hyperbolic $3$-manifold. Are there two, nonhomotopic, $\pi_1$-injective closed surfaces in $M$, and which do not intersect?
- 553
4
votes
1
answer
317
views
Immersed incompressible surfaces in surface bundles
Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not ...
- 553
7
votes
1
answer
475
views
Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?
I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.
Following this paper by Christian ...
- 318
6
votes
3
answers
555
views
Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
- 3,751
3
votes
2
answers
950
views
hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
...
- 1,629
2
votes
1
answer
183
views
Shearing in hyperbolic 3-manifolds
I'm new to 3-manifolds, and while reading an article (arXiv link) by Hongbin Sun about virtual domination of hyperbolic manifolds, I got a little bit confused, he says about $1+\pi i$-shearing (page ...
49
votes
3
answers
8k
views
Thurston's 24 questions: All settled?
Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":
$\cdots$
Two naive questions from an outsider:
(1) Have all $24$ now been resolved?
(2)...
- 151k
6
votes
1
answer
2k
views
Definition of cusped manifold?
There is much talk about hyperbolic cusped 3-manifolds, but almost no definition of what a cusped manifold is.
One definition I found was that it is a result of a parabolic transformation on H^n, ...
- 1,465
2
votes
2
answers
302
views
Complement of figure 8 knot - zero vertex [closed]
Thurston in "3-dimensional Geometry and Topology" explicitely creates the hyperbolic complement of the figure 8 knot by glueing two ideal tetrahedra.
What I do not understand is that he pushes the ...
- 1,465
5
votes
1
answer
282
views
Non-Haken hyperbolic 3-manifolds without nonorientable surfaces
It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...
- 553
10
votes
0
answers
127
views
Compatibility of spherical and hyperbolic geometry for fibred knots
Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
- 44.4k
11
votes
3
answers
821
views
Random links and $3$-manifolds
In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ ...
- 1,314
8
votes
1
answer
292
views
Conditions on the hierarchy for Thurston's hyperbolization theorem
From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, $\partial$-...
- 367
6
votes
2
answers
294
views
Heegard genus of hyperbolic Haken 3-manifolds
Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?
- 553
3
votes
1
answer
375
views
Chern-Simons invariants of 2-bridge knots
2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below:
Looking at some ...
- 10.4k
9
votes
2
answers
812
views
How many knots are there with hyperbolic volume less than a given constant
Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...
- 18.7k
9
votes
4
answers
1k
views
Geometry of the space of circles in the Euclidean plane
We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.
It may even be possible to write an explicit formula ...
- 22.8k
10
votes
2
answers
650
views
Heegaard genera of arithmetic 3-manifolds
UPDATE: Because I was hoping that state the question as concisely as
possible, the original post did not include a precise definition of
arithmetic 3-manifold only a reference to Maclachlan and ...
- 5,259
8
votes
1
answer
613
views
Virtual fibering conjecture for cusped hyperbolic manifolds
I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the ...
6
votes
1
answer
800
views
Geometrization & JSJ decomposition with boundary
Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
- 125
0
votes
1
answer
291
views
A question on Cayley graphs and hyperbolic 3-manifolds
There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...
- 841