All Questions
Tagged with hyperbolic-geometry reference-request
83 questions
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
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
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
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
1
vote
1
answer
266
views
Reference for the result that the systol map from Teichmuller space to curve complex is coarsely Lipschitz
Let $\mathcal{T}(S)$ denotes the Teichmuller space of a finite type surface $S$ equipped with Teichmuller metric and $\mathcal{C}(S)$ denotes the curve complex. Define a map $$\phi:\mathcal{T}(S)\...
- 1,713
1
vote
2
answers
360
views
Does the Teichmüller space of the pair of pants admit a continuous global section?
Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth ...
- 4,304
7
votes
1
answer
343
views
For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?
The fractal dimension of the 3D Apollonian packing is computed in this paper.
In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...
- 2,581
1
vote
1
answer
271
views
Ratner theorem and dense geodesic planes in hyperbolic manifolds
Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
- 829
4
votes
2
answers
479
views
Visibility spaces and Gromov hyperbolicity
I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...
- 345
3
votes
2
answers
246
views
Conformal invariants of planar pairs of pants
Consider a planar pair of pants
$$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r_1, |z+x| \ge r_2 \right\rbrace$$
where $-1 < -x-r_2 < -x+r_2 < 0 < x-r_1 < x+r_1 < 1$.
...
- 4,304
9
votes
0
answers
359
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
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
0
votes
1
answer
153
views
Entropy of Negatively pinched manifolds
Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...
- 1,101
18
votes
2
answers
4k
views
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
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
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
0
votes
0
answers
177
views
Existence of special pants decompositions for non-elementary representations into PSL(2,R)
A Theorem by Gallo, Goldman and Porter states the following:
Let $S_g$ be a closed orientable surface of genus $g$ with fundamental group $\Gamma_g$, and fix a non-elementary representation $\rho\...
- 3,749
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
7
votes
0
answers
1k
views
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
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
0
votes
0
answers
313
views
Quick references/sources for the hyperbolic Riemann Surfaces with boundary
Hello,
Here I am asking for a reference for the universal cover of hyperbolic Riemann surfaces with geodesic boundaries. For example, I want to know how the universal cover/fundamental domain of ...
- 1,471
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
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
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
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
1
vote
1
answer
294
views
Good references for Hyperbolic and parabolic annuli
I want to understand the geometry of the Hyperbolic annuli (Hyperbolic plane quoteinted by the group generated by $z\mapsto rz$ for a fixed $r$) and parabolic annuli (Hyperbolic plane quoteinted by ...
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
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
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
2
votes
3
answers
1k
views
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,
...
- 3,749
19
votes
1
answer
901
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