All Questions
22 questions
3
votes
0
answers
463
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
- 613
5
votes
1
answer
291
views
Question about and good reference for Kahn and Markovic result
As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:
Let $M$ be a compact hyperbolic $3$...
- 829
3
votes
1
answer
375
views
Reference for triangle groups
Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
- 613
2
votes
1
answer
205
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Is every finitely generated classical Schottky group quasifuchsian?
$\DeclareMathOperator\PSL{PSL}$(Classical, finitely generated) Schottky groups are groups generated by finitely many hyperbolic elements of $A_i\in \PSL(2,\mathbb{C}), $ $i<n$ such that the ...
- 441
1
vote
2
answers
336
views
The moduli space of finite volume hyperbolic 3-manifolds?
By finite volume hyperbolic 3-manifold, I do mean $M=\mathbb{H}^{3}/\Gamma$ where $\Gamma$ is a torsion-free Kleinian group such that the hyperbolic volume $Vol(M)<\infty$.
I will call
$$\mathcal{M}...
- 223
3
votes
1
answer
173
views
Cusps of hyperbolic surfaces under finite covers
The following statement seems true, but I don't know a proof or a reference for it (and I would like one).
Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We ...
- 291
4
votes
0
answers
88
views
What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?
Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
- 2,533
2
votes
2
answers
330
views
$PSL_2(\mathbb{R})$ representations of free groups
Let $S_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S_{g,n}^b$ deformation retracts to a bouquet ...
- 445
5
votes
1
answer
142
views
Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
- 14.3k
5
votes
1
answer
394
views
closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface
Could you please recommend me some references for proofs of this fact: "closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface". Thanks in advance!
- 333
18
votes
2
answers
1k
views
Hyperbolic Volume and Chern-Simons
In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons ...
- 183
3
votes
2
answers
950
views
hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
...
- 1,629
8
votes
2
answers
631
views
Teichmüller space on non-orientable closed surfaces
It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...
- 81
11
votes
3
answers
822
views
Random links and $3$-manifolds
In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ ...
- 1,314
2
votes
3
answers
511
views
Blaschke Condition for hyperbolic lattices
For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
- 975
24
votes
2
answers
1k
views
SnapPea for the uninitiated
SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the ...
- 56.9k
2
votes
2
answers
487
views
Some general properties of arithmetic groups of simplest type
I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...
- 2,436
1
vote
1
answer
271
views
Ratner theorem and dense geodesic planes in hyperbolic manifolds
Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
- 829
5
votes
3
answers
593
views
Who first used the cross-ratio to describe shapes in hyperbolic geometry?
I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes
and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...
- 3,359
6
votes
1
answer
800
views
Geometrization & JSJ decomposition with boundary
Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
- 125
5
votes
1
answer
439
views
Rotation part of short geodesics in hyperbolic mapping tori
Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
- 1,601
19
votes
1
answer
902
views
Locus of equal area hyperbolic triangles
Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
- 565