52
votes
Accepted
Thurston's 24 questions: All settled?
A nice summary of the status of these problems may be found here:
Otal, Jean-Pierre, William P. Thurston: ``Three-dimensional manifolds, Kleinian groups and hyperbolic geometry", Jahresber. Dtsch. ...
- 61.2k
37
votes
Accepted
Interactive model of the hyperbolic plane for a general public lecture
By chance I wrote, not long ago, the following applet (HTML5+JS+WebGL) that works at least on Firefox and Chrome.
https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Escher/
This work is CC-BY-...
- 1,497
32
votes
Smallest tile to tessellate the hyperbolic plane
Binary Tiling
In fact, one can tile the hyperbolic plane with arbitrarily small tiles. There is a tiling of the hyperbolic plane (apparently due to Boroczky) by pentagons.
The horizontal edges are ...
- 61.2k
32
votes
Accepted
Immersions of the hyperbolic plane
Yes, it immerses isometrically into certain solvmanifolds. Take an Anosov map of $T^2$, such as $\left[\begin{array}{cc}2 & 1 \\1 & 1\end{array}\right]$. The mapping torus admits a locally ...
- 61.2k
24
votes
Interactive model of the hyperbolic plane for a general public lecture
Live isometries of a hyperbolic ornament
As part of my dissertation Creating Hyperbolic Ornaments I wrote some Java software which might serve your needs. It's called morenaments conform (as of now). ...
- 534
23
votes
Accepted
Isometry group of a compact hyperbolic surface
In genus 2, every surface has a symmetry, namely a hyperelliptic involution. In higher genus, generic surfaces will not have any symmetries. If a surface has a non-trivial symmetry group, then the ...
- 61.2k
23
votes
Thurston's 24 questions: All settled?
They have all been resolved in some fashion with the exception of problem 23, which asserts that the volumes of all hyperbolic 3-manifolds are not rationally related. We know basically nothing about ...
- 39.8k
21
votes
Accepted
For which surfaces is Penner's conjecture known to be true?
Shin and Strenner have shown that the conjecture is false when 3g + n > 4.
See http://arxiv.org/abs/1410.6974
21
votes
Accepted
Geometry of the space of circles in the Euclidean plane
Things will simplify if you just consider the circles on the Riemann sphere $S^2 = \mathbb{C}\cup\{\infty\}$, for your space is simply the space of circles on the sphere (with the lines in $\mathbb{C}$...
- 97.6k
21
votes
Why are Fuchsian groups interesting?
About the relation to fractals: for Fuchsian groups of the first kind, the limit set has Hausdorff dimension 1, i.e., it is not fractal.
However, for all other quasifuchsian groups of the first kind,...
- 9,786
19
votes
Accepted
Random links and $3$-manifolds
There is no universally accepted model of random three-manifolds (or random knots/links) for that matter, however, hyperbolicity is pervasive in all known models. The most popular (but not really ...
- 93.7k
19
votes
Accepted
Smallest tile to tessellate the hyperbolic plane
The tilings mentioned by Ian Agol are related to an action of a Baumslag-Solitar group $\{ a,b \bigg| b^{-1}a^2b=a \}$ on the hyperbolic plane. They have arbitrarily small area, but diameter ...
- 27.1k
19
votes
Accepted
Hyperbolic $3$-manifold groups that embed in compact Lie groups
All closed hyperbolic 3-manifold groups embed into a compact Lie group.
To prove this, note first of all that given a hyperbolic 3-manifold $M$, it suffices to show that a finite-index subgroup $G\...
- 61.2k
18
votes
Accepted
The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions
I think Scott's argument is that the lengths of $6g-6$ curves can't form coordinates for Teichmuller space. If one has $6g-6$ geodesics which parameterize, then they must be filling (they meet every ...
- 61.2k
18
votes
17
votes
Accepted
Does Helly's theorem hold in the hyperbolic plane?
I don't understand your reference to the model (since the geometry of the hyperbolic plane does not depend on any model), but, in fact, the Beltrami-Klein model demonstrates that any qualitative ...
- 93.7k
16
votes
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
I wondered this myself, I made some similar pictures to approximate the stable lamination of a pseudo-Anosov map (the blue curve is a geodesic, the other colors horocycles).
Every geodesic on the ...
- 61.2k
15
votes
Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
The simplest answer is that the universal cover together with the deck group action contain a lot of information about the manifold, and the representations of the group provide one way to ...
- 32.2k
15
votes
Accepted
Pseudo-Anosov maps with same dilatation.
Yes. If two pseudo-Anosov mapping classes are conjugate then they must have the same dilatation. So take any pseudo-Anosov $f$ and any mapping class $h$ not in the centraliser of $f$ and let $g = h f ...
- 3,055
14
votes
Accepted
Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$
I do not know of any specific reference, but I will try to explain the situation a bit. For most of the stuff, one does not actually need references besides the paper of Borel-Serre defining the Borel-...
- 16.5k
14
votes
Accepted
Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?
Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the ...
- 24.4k
14
votes
Accepted
Besides the tracioid are there other surfaces of revolution that have a constant negative curvature?
There are many examples of surfaces in $\mathbb{R}^3$ with constant negative curvature. They can be described by using the so-called parametrization by Chebyshev nets. Have a look at the paper by ...
- 61.9k
14
votes
Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Three manifold groups are quite special and their representation varieties have more structure than that of a random group. Casson's view point is helpful to see the point. The Heegaard decompoistion ...
- 2,894
14
votes
Interactive model of the hyperbolic plane for a general public lecture
Not exactly what you ask, but there is a computer game HyperRogue taking place on hyperbolic plane. Having an actual protagonist moving around the plane is an excellent way to introduce hyperbolic ...
- 7,296
14
votes
Accepted
Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?
The results proved in
S. Kojima, "Deformations of hyperbolic 3-cone-manifolds",
J. Differential Geom. 49 (1998), no. 3, 469-516
provide complete answers to questions 1 and 3.
The main theorem of ...
- 3,619
13
votes
Accepted
Failure of Mostow rigidity in dimension 2
ad (i):
First consider a Dehn twist at some simple, closed curve in a closed hyperbolic surface. It is obviously a quasi-isometry (as any smooth map between closed surfaces) but not an isometry.
...
- 9,786
13
votes
Accepted
How many knots are there with hyperbolic volume less than a given constant
Here is an expansion of Ian's answer.
Dehn surgery on a hyperbolic knot or link generically gives another hyperbolic manifold. This follows from the Dehn surgery theorem; see Theorem 5.8.2 in ...
- 18.7k
13
votes
Isometry group of a compact hyperbolic surface
I'll just point out an answer based on somewhat different criteria than explicitly knowing features of the hyperbolic metric: As is well-known, every oriented, compact hyperbolic surface $C$ is ...
- 97.6k
13
votes
Accepted
A geometric proof of the Gauss-Lucas theorem
Good question! I think the following article may qualify as a "yes":
Arnaud Ch´eritat, Yan Gao, Yafei Ou, Lei Tan. "A refinement of the Gauss-Lucas theorem (after
W. P. Thurston)". 2015.
https://hal....
- 2,005
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