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. ...

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-...

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 ...

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 ...

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). ...

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 ...

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 ...

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

Community wiki

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 $\mathbb{C}$...

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,...

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 ...

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 ...

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\...

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 ...

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 ...

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 ...

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 ...

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 ...

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-...

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 ...

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 ...

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 ...

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 ...

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 ...

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.
...

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 ...

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 ...

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....

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