All Questions
Tagged with hyperbolic-geometry reference-request
83 questions
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
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 ...
- 28.9k
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
18
votes
2
answers
4k
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Reference request: Geodesic flow on a manifold with negative curvature is ergodic
I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...
- 957
18
votes
2
answers
1k
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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
13
votes
3
answers
2k
views
Isometry group of a compact hyperbolic surface
Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-...
- 133
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 ...
- 481
11
votes
3
answers
823
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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
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 (...
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
10
votes
1
answer
579
views
Periodic billiard paths in hyperbolic triangles
It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path,
one obeying the reflection law that the angle of incidence equals the angle of reflection. ...
- 151k
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,...
- 1,471
9
votes
1
answer
480
views
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 (...
- 18.7k
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
9
votes
0
answers
360
views
Phillips-Sarnak conjecture in higher dimension
The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...
- 245
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
7
votes
1
answer
957
views
Relations between some works by Deligne-Mostow and Thurston
Happy new year 2016!
A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...
- 347
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 ...
- 71
7
votes
1
answer
343
views
For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?
The fractal dimension of the 3D Apollonian packing is computed in this paper.
In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...
- 2,581
7
votes
0
answers
1k
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Closed geodesics on a closed, negatively curved Riemannian manifold
I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...
- 71
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
6
votes
1
answer
414
views
Uniformization of a plane minus cantor set
Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set.
...
- 4,304
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 ...
- 63
6
votes
1
answer
338
views
Reference request: embedding the hyperbolic triangulation in $\mathbb{R}^3$
Let $T_d$ be the infinite valence $d$ triangulation of the hyperbolic plane, where each triangle is equilateral and $d \ge 7$. Question: Is there an isometric embedding from $T_d \to \mathbb{R}^3$? ...
- 1,798
6
votes
1
answer
434
views
Previous work on this generalization of continued fractions?
The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...
- 1,532
6
votes
0
answers
200
views
Spectral theory for Dirac Laplacian on a funnel
I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
- 313
5
votes
3
answers
432
views
What is the limit set of a hyperbolic lattice?
My claim is as follows:
Let $\Gamma$ be a discrete subgroup of $\operatorname{Isom}(\Bbb{H}^{n})$, the isometries of hyperbolic $n$-space. If $\Gamma$ is a lattice in $\operatorname{Isom}(\Bbb{H}^n)...
- 195
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 ...
- 445
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
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
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
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
241
views
Uniqueness of hyperbolic rescaling
Let $X$ be a compact oriented surface of genus at least two, equipped with a Riemannian metric $g$. By the uniformization theorem for Riemann surfaces, there is a conformal universal covering map $p:...
- 56.9k
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
5
votes
1
answer
353
views
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} &...
- 6,548
4
votes
2
answers
479
views
Visibility spaces and Gromov hyperbolicity
I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...
- 345
4
votes
2
answers
998
views
A construction of generators of discrete subgroups of SL(2,R)
I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...
- 211
4
votes
1
answer
198
views
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 ...
- 6,025
4
votes
1
answer
307
views
The Weyl law for lengths
For what I know, this must be a standard fact, but I can't spot it in the literature I have on hands. What is the asymptotic of the geodesic
lengths spectrum for the modular surface $X(1)$? (That is, ...
- 6,891
4
votes
2
answers
378
views
Comparing two Delaunay tessellations on a hyperbolic surface
Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb H\...
- 838
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 ...
- 81
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
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
4
votes
0
answers
670
views
Thales' Theorem for Hyperbolic Geometry [duplicate]
In Euclidean geometry Thales' Theorem says that if you view a diameter of a circle from any point on the perimeter it occupies exactly $90$ degrees in your field of view.
More generally for any ...
- 4,304
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
3
votes
3
answers
1k
views
Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry
Hello,
Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also ...
- 1,471
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 (...
- 31
3
votes
1
answer
158
views
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 ...
- 1,804
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