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. ...
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  • 61.2k
37 votes
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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-...
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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 ...
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  • 61.2k
32 votes
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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 ...
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  • 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). ...
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  • 534
23 votes
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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 ...
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  • 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 ...
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  • 39.8k
21 votes
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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
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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 $\mathbb{C}$...
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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,...
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  • 9,786
19 votes
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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 ...
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  • 93.7k
19 votes
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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 ...
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  • 27.1k
19 votes
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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\...
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  • 61.2k
18 votes
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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 ...
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  • 61.2k
18 votes

Why are Fuchsian groups interesting?

Check out Indra's Pearls. (Mumford, Series, Wright).
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  • 93.7k
17 votes
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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 ...
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  • 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 ...
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  • 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 ...
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15 votes
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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 ...
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  • 3,055
14 votes
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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-...
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14 votes
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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 ...
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  • 24.4k
14 votes
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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 ...
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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 ...
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  • 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 ...
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  • 7,296
14 votes
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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 ...
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13 votes
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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. ...
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  • 9,786
13 votes
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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 ...
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  • 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 ...
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13 votes
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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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