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. ...

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-...

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). ...

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 ...

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 ...

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

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}(...

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

### 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 ...

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 ...

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 ...

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

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

### 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....

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 ...

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 ...

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 ...

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; ...

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) ...

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

hyperbolic-geometry × 837gt.geometric-topology × 242

dg.differential-geometry × 133

mg.metric-geometry × 107

riemannian-geometry × 92

3-manifolds × 89

reference-request × 84

gr.group-theory × 82

riemann-surfaces × 78

teichmuller-theory × 77

geometric-group-theory × 76

knot-theory × 35

cv.complex-variables × 29

fuchsian-groups × 29

ag.algebraic-geometry × 28

complex-geometry × 28

ds.dynamical-systems × 23

at.algebraic-topology × 20

lie-groups × 20

moduli-spaces × 20

rt.representation-theory × 18

nt.number-theory × 16

mapping-class-groups × 16

surfaces × 16

ergodic-theory × 15