All Questions
Tagged with hyperbolic-geometry gt.geometric-topology
264 questions
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
40
votes
1
answer
1k
views
Four circles on the sphere
Consider generic configurations consisting of 4 distinct circles on the sphere.
Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to ...
- 91.8k
31
votes
4
answers
2k
views
Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
- 22.1k
30
votes
7
answers
5k
views
Why are Fuchsian groups interesting?
I keep hearing that fuchsian groups are interesting for other reasons than the Fuchsian model for hyperbolic Riemann surfaces.
What are those reasons?
Are the Fuchsian groups with fixed points ...
- 893
24
votes
2
answers
1k
views
SnapPea for the uninitiated
SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the ...
- 56.9k
20
votes
2
answers
2k
views
Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
- 6,353
19
votes
1
answer
901
views
Locus of equal area hyperbolic triangles
Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
- 565
18
votes
3
answers
1k
views
How to see isometries of figure 8 knot complement
The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
- 518
18
votes
3
answers
1k
views
The number of cusps of higher-dimensional hyperbolic manifolds
Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp.
Could ...
- 3,749
18
votes
2
answers
1k
views
Hyperbolic Volume and Chern-Simons
In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons ...
- 183
18
votes
2
answers
2k
views
How does hyperbolicity of space time affect our lives?
My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold.
But recently I found ...
- 3,359
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
17
votes
1
answer
1k
views
Hyperbolic manifolds which fiber over the circle
If $N^2$ is a closed, orientable surface of genus at least $2$, and if $\phi$ is an (orientation-preserving) pseudo-Anosov mapping on $N$, then one can form the closed orientable 3-manifold $M^3$ by ...
- 4,425
15
votes
1
answer
359
views
Are hyperbolic $n$-manifolds recursively enumerable?
Fixing a dimension $n \ge 4$, is the class of closed hyperbolic $n$-manifolds recursively enumerable?
Since hyperbolic manifolds are triangulable I can reformulate this in the following more explicit ...
- 3,399
15
votes
1
answer
454
views
Bloch group, hyperbolic manifolds and rigidity
I have some questions concerning the hyperbolic geometry side of the rigidity question for $K_3$ which asks if the natural map $K_3^{\operatorname{ind}}(\overline{\mathbb{Q}})\to K_3^{\operatorname{...
- 17.4k
14
votes
2
answers
617
views
For which surfaces is Penner's conjecture known to be true?
Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...
- 540
14
votes
1
answer
908
views
Comparing layered triangulations of 3-manifolds which fiber over the circle.
I am sorry but I am reposting this question because I wasn't logged in when I first asked it.
Ian Agol has produced a method to build an ideal layered triangulation of a hyperbolic 3-manifold which ...
- 540
13
votes
3
answers
1k
views
Flat SU(2) bundles over hyperbolic 3-manifolds
Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold?
The literature on such bundles over 3-manifolds is huge and my naive searches don'...
- 6,247
13
votes
2
answers
1k
views
Closed hyperbolic manifold with right-angled fundamental domain
What is an example (as simple as possible, please!) of a closed hyperbolic three-manifold with a right-angled polyhedron as fundamental domain?
If we allow cusps then the Whitehead link or the ...
- 28.2k
13
votes
1
answer
2k
views
Detecting a cover of the figure-8 knot complement
I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this ...
- 418
13
votes
3
answers
1k
views
Best known Margulis constants?
A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...
- 1,601
13
votes
1
answer
1k
views
Topology of boundaries of hyperbolic groups
For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...
- 305
13
votes
2
answers
485
views
Geodesic current supported on a pencil?
Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...
- 10.1k
13
votes
1
answer
1k
views
Representation varieties of 3-manifold groups in $\mathrm{SL}(n,\mathbb{C})$
I am looking at the variety of representations of the fundamental group of a hyperbolic 3-manifold into $\mathrm{SL}(n,\mathbb{C})$:
$$\mathrm{Hom}(\pi_1(M), \mathrm{SL}(n,{\mathbb C}))$$
It is known ...
- 10.4k
13
votes
0
answers
371
views
What is the cup-product structure like on a hyperbolic 5-manifold?
Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?
For example, are there hyperbolic 5-manifolds ...
- 4,949
12
votes
2
answers
2k
views
The work of Thurston
I seem to remember written or said somewhere that at some point Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field (maybe this is not correct?). I ...
- 28.9k
12
votes
3
answers
1k
views
Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?
Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...
- 10.5k
12
votes
1
answer
797
views
Finite covers of hyperbolic surfaces and the `second systole´
We are interested in the following ´relative´ version of residual finiteness for fundamental groups of surfaces. Similar discussions where given in this question: injectivity radius of hyperbolic ...
- 3,928
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
11
votes
2
answers
1k
views
Is there a contractible hyperbolic 3-orbifold of finite volume?
Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...
- 423
11
votes
2
answers
811
views
Can you cover a genus a billion hyperbolic surface with 15 balls?
Here's a question I was wondering about this week. Not sure how interesting it is, but I thought it was kind of curious.
Question: Given $k$, is there a number $N=N(k)$ such that if a closed ...
- 532
11
votes
2
answers
620
views
Some mid-sized ¿hyperbolic? manifolds and SnapPea
I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question
can you fool SnapPea?
but in ...
- 44.4k
11
votes
1
answer
332
views
Rank and hyperbolic volume
Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of ...
- 96.4k
11
votes
1
answer
902
views
Why isn't $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}\rangle$ a hyperbolic surface group?
The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed torus fundamental ...
- 3,359
11
votes
0
answers
352
views
Right-angled polytopes
%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not exist ...
- 1,268
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
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
10
votes
1
answer
568
views
Is the spectrum of closed geodesics in a closed hyperbolic 3-manifold asymptotically homogeneously dense?
It seems to me that the results in this important paper of Kahn-Markovich imply the following fact. Let $M^3$ be any closed hyperbolic 3-manifold. For every $\epsilon > 0$ there is a natural number ...
- 10.5k
10
votes
2
answers
488
views
Question about the Weeks Manifold
I am wondering about the existence of incompressible surfaces in the Weeks manifold. Is this space a Haken manifold?
- 553
10
votes
1
answer
501
views
Examples of 3-manifolds with RFRS fundamental group
I'm wondering if anyone knows how to construct hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in ...
- 1,329
10
votes
1
answer
737
views
Parabolic subgroups of relatively hyperbolic and CAT(0) groups
Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...
- 2,090
10
votes
1
answer
2k
views
Questions on Thurston's earthquake flow
$\DeclareMathOperator\PSL{PSL}$Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references.
I briefly recall the settings. Let's fix a closed ...
- 1,804
10
votes
1
answer
416
views
Diameter of hyperbolic 3-manifolds
Is there a good method to estimate the diameter of a closed hyperbolic 3-manifold?
I am particularly interested in know the diameter of the Weeks manifold.
- 553
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
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
9
votes
3
answers
801
views
What constant ensures hyperbolicity of Dehn surgery?
I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...
- 1,000
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
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
9
votes
2
answers
691
views
Do different Dehn fillings produce homeomorphic 3-manifolds ?
Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold.
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a
hyperbolic structure. ...
- 841
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