All Questions
Tagged with hyperbolic-geometry reference-request
83 questions
12
votes
4
answers
991
views
Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space
Consider the group $\operatorname{PSL}(2,\mathbb C)$ acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is ...
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 ...
- 28.2k
0
votes
2
answers
118
views
Why is a plane graph Delaunay realizable if stellating a face makes the graph inscribable?
I have come across the following lemma in several papers (for instance see lemma 2.2 in this) (and some authors state this the proof immediately follows from basic properties of the stereographic ...
3
votes
1
answer
274
views
Counterexample to mostow rigidity theorem
I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...
- 44.4k
4
votes
1
answer
198
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Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces
I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
- 12.3k
2
votes
0
answers
60
views
The relationship between convex hulls
Consider a (f.g., classical) Schottky group acting on $\mathbb H^3$; consider a convex hull of the limit set $C(\Lambda)$ and a convex hull of a closure of an orbit of a point on $\mathbb CP^1,$ $C(\...
- 441
9
votes
1
answer
480
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What is the complex structure on the boundary torus of a hyperbolic knot complement?
Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary (...
- 1,446
2
votes
0
answers
130
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Double quotient integral formula on $\Gamma \backslash G /K$
Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...
- 457
2
votes
0
answers
109
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Further directions in representations of surface group into a Lie group
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.
Now I am planning to ...
CommunityBot
- 1
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 $3$...
- 674
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,...
- 63.9k
10
votes
4
answers
667
views
Reference for shortest educational path to (Riemannian) hyperbolic plane
I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
- 2,218
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.
- 12.9k
3
votes
1
answer
165
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Reference request for statement concerning free subgroups of $ \mathrm{SL}_2(\mathbb{Z}). $
I am interested in finding a reference for the following claim:
There exists a free subgroup $F_2$ of $\mathrm{SL}_2(\mathbb{Z})$ on two generators that does not contain any nontrivial unipotent ...
- 28.2k
3
votes
0
answers
168
views
Variants of Selberg trace formula
I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...
- 31
2
votes
1
answer
205
views
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 ...
- 63.9k
4
votes
0
answers
433
views
Convex core and geometric finiteness of negatively curved manifolds
I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...
- 41
5
votes
1
answer
439
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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 ...
- 35.5k
7
votes
3
answers
413
views
Best source for classification of right-angled hyperbolic hexagons
A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...
- 18.7k
4
votes
1
answer
283
views
Mapping the hyperbolic plane onto the interior of a disk
In Chapter II of his book Non-Euclidean Geometry (1961; first published in Polish, 1956), Stefan Kulczycki defines a mapping of the hyperbolic plane onto the interior of a disk. Its construction ...
- 428
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}...
- 28.2k
3
votes
1
answer
142
views
Busemann-Feller lemma in hyperbolic space
The classical Busemann-Feller lemma in Euclidean space says the following.
Let $K\subset \mathbb{R}^n$ be a closed convex set.
Then
for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...
- 21.8k
3
votes
2
answers
420
views
Learning roadmap for Lorentzian geometry
I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF.
I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (...
- 7,817
2
votes
0
answers
85
views
Showing an action of a higher rank lattice on hyperbolic space has a fixed point
In the introduction to this paper, the author mentions that any action of a lattice $\Gamma < G$ on a rank one symmetric space $X$ has a fixed point, where $G$ is a higher rank semisimple algebraic ...
- 191
2
votes
2
answers
313
views
References on Riemann surfaces
I have asked the question in MSE, but did not get an answer.
I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...
- 1,594
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 ...
- 15.4k
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
24
votes
3
answers
1k
views
Hyperbolic Coxeter polytopes and Del-Pezzo surfaces
Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf
I would like to find a reference for a beautiful ...
- 2,175
9
votes
0
answers
331
views
Connections between spectral geometry and critical point/Morse theory
I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
- 217
1
vote
0
answers
42
views
Effect of plumbing a surface on the marked length spectrum
First I'll recall the plumbing procedure.
Let $M$ be a noded Riemann surface with nodes $p_1,\dots, p_n$. There is a family of pairwise disjoint neighbourhoods of each node $U_i$ that has coordinates ...
- 445
1
vote
0
answers
92
views
Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind
Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...
- 333
11
votes
4
answers
649
views
Introductory textbook on geometry of hyperbolic space
I am looking for an introductory textbook to the geometry of the hyperbolic space $\mathbb{H}^n$. The book should include explicit description of geodesics and horospheres in various models (...
- 1,181
5
votes
3
answers
295
views
Teichmuller space for surface with cone points
Working on my current research problem, Teichmuller spaces for surfaces with cone points have come into play. It's fairly easy to formulate some of the definitions, a few basic results, etc. To be ...
- 91.8k
1
vote
0
answers
1k
views
Geodesic curvature in hyperbolic geometry- Poincaré disk model [closed]
How is geodesic curvature defined with an ODE or geometric construction in the hyperbolic plane?
How do hyperbolic geodesics change when there is deviation from geodesy? How to construct or graph ...
- 917
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 ...
- 8,529
1
vote
1
answer
393
views
Hyperbolic embedding of a directed acyclic graph defined over strings
For integer $n$ and alphabet $\Sigma$ we construct a DAG (directed acyclic graph) $G=(V,E)$ over strings $s\in\Sigma^\star$ as follows:
$$V = \{s\in\Sigma^\star\colon |s|\le n\}$$
$$E = \{(s_1,s_2)\...
- 187
6
votes
1
answer
157
views
Configurations of $n$ points modulo isometries of the ambient space
Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate ...
- 28.9k
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 ...
- 5,259
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 $\{r,s\}$ ...
- 349
3
votes
1
answer
158
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Reference request: geometric finiteness of Fuchsian groups
My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong):
Proposition. The convex core of a complete hyperbolic surface ...
- 993
3
votes
0
answers
349
views
The uniqueness of Poincaré metric
The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance.
...
- 7,194
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 ...
- 68.9k
5
votes
1
answer
394
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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!
CommunityBot
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2
votes
0
answers
66
views
A boundary for integrals of eigenfunctions over geodesics?
Let $X$ be a compact hyperbolic surface, and $\gamma$ a closed geodesic on it.
Consider the integral
$$\int_\gamma f(x)\, dl(x)$$
where $f$ is a (normalized) Laplace eigenfunction on $X$. ...
- 6,891
9
votes
6
answers
4k
views
Books for hyperbolic geometry ( surfaces ) with exercises?
what are good books on hyperbolic geometry/hyperbolic surfaces that have good number of exercises, just to get a good understanding of the literature . I know John Ratcliffe's book will be one of them,...
- 4,713
2
votes
3
answers
1k
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Reference for the geometry of horospheres
I am looking for a reference to a proof of the following well-know fact (cited for example by
B.Farb in ``Relatively hyperbolic groups'', Geom. Funct. Anal. 8 (1998), no. 5, 810--840); MR1650094,
...
- 4,713
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 ...
- 10.4k
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?
...
- 68.9k
3
votes
0
answers
322
views
Curvature $\geq-1$ but not $\geq1$
(Edited again)
In the following, for brevity, I will say that $$X\ \ \mathrm{has}\ \ \kappa_{\mathrm{max}}=k$$ if $X$ is a compact ($n$-dimensional with $n\geq2$, with empty boundary) Aleksandrov ...
CommunityBot
- 1
5
votes
1
answer
353
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Quantifying the monotonicity property of the hyperbolic metric
Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
CommunityBot
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