53
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. ...
- 66k
38
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-...
- 1,507
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 ...
- 66k
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 ...
- 66k
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
24
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 ...
- 42.4k
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 ...
- 66k
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,...
- 10.2k
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 ...
- 95.2k
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 ...
- 27.8k
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\...
- 66k
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 ...
- 66k
18
votes
18
votes
Accepted
On trivial mapping class group of 3-manifolds
Dave Gabai proved that the mapping class group of a closed hyperbolic 3-manifold is isomorphic to its isometry group. For a hyperbolic knot $K$ without any symmetries, for large enough $n$, $S^3_{1/n}(...
- 66k
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 ...
- 95.2k
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 ...
- 66k
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 ...
- 33.5k
15
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 ...
- 7,696
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 ...
- 3,115
15
votes
Accepted
Is there a contractible hyperbolic 3-orbifold of finite volume?
Yes. For example, let $M$ be the figure-eight knot complement. So $M$ is a hyperbolic manifold with volume a bit more than 2. The manifold $M$ has a two-fold symmetry $\tau$ that fixes, pointwise, a ...
- 24.2k
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 ...
- 2,914
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 ...
- 3,699
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 ...
- 105k
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....
- 2,115
13
votes
Why are Fuchsian groups interesting?
Fuchsian groups occur naturally in JSJ-theory. In 3-manifolds they occur as the base Groups of the Seifert pieces and in geometric group theory they occur as (quotients of) enclosing groups (also ...
- 446
13
votes
Why are Fuchsian groups interesting?
Fuchsian groups, particularly those which are cocompact, form the tips of several big mathematical icebergs. To put this less metaphorically, several discoveries about Fuchsian groups, obtained using ...
- 15.2k
13
votes
Surfaces with non-constant negative curvature
If you just want examples for which it's not hard to figure out how the geodesics behave, here's a class of examples with negative and non-constant curvauture in the plane where the geodesics are ...
- 105k
13
votes
Accepted
Is Tarskian hyperbolic geometry consistent, complete & decidable?
The canonical reference for Tarski-style elementary geometry is the monograph Schwabhäuser, Szmielew, Tarski [1]. This includes a treatment of hyperbolic geometry in parallel with Euclidean geometry; ...
- 43k
12
votes
Harmonic spinors on closed hyperbolic manifolds
I'm happy to be able to answer my own question! John Ratcliffe, Steven Tschantz and I showed that the Dirac operator on the Davis manifold (a closed hyperbolic 4-manifold constructed by Mike Davis) ...
- 18.9k
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