# Tag Info

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
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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-...
• 1,497

### 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
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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 ...
• 61.2k

### 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
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

### 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
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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
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

### 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
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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 ...
• 93.7k
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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 ...
• 27.1k
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• 3,055
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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-...
• 16.5k
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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 ...
• 24.4k
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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 ...
• 61.9k

### 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

### 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
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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 ...
• 3,619
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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. ...
• 9,786
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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 ...
• 18.7k

### 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