All Questions
Tagged with hyperbolic-geometry reference-request
83 questions
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 ...
- 7,831
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)...
- 68.9k
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 ...
- 63.9k
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. ...
- 96.4k
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, ...
- 3,399
2
votes
0
answers
78
views
automorphic forms associated with symmetries of vertices of uniform honeycombs in hyperbolic space
Is there a catalogue of automorphic forms (modular/Maass/Siegel/Hilbert...) which lists them in terms of Poincaré series associated with the symmetries of the vertices of uniform honeycombs in ...
- 975
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
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 $M_1\# M_2$ ...
CommunityBot
- 1
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
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-...
- 91.8k
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 ...
- 5,259
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.
...
- 4,131
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 (...
CommunityBot
- 1
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 $$\phi:\mathcal{T}(S)\...
- 28.2k
4
votes
2
answers
479
views
Visibility spaces and Gromov hyperbolicity
I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...
- 51
1
vote
1
answer
271
views
Ratner theorem and dense geodesic planes in hyperbolic manifolds
Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
- 17k
1
vote
2
answers
360
views
Does the Teichmüller space of the pair of pants admit a continuous global section?
Let $P$ be a pair of pants, $H(P)$ be the space of smooth hyperbolic Riemannian metrics with geodesic boundary on $P$, and $T(P)$ be the Teichmüller space of $P$ (quotient of $H(P)$ under smooth ...
- 31.2k
3
votes
2
answers
246
views
Conformal invariants of planar pairs of pants
Consider a planar pair of pants
$$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r_1, |z+x| \ge r_2 \right\rbrace$$
where $-1 < -x-r_2 < -x+r_2 < 0 < x-r_1 < x+r_1 < 1$.
...
- 96.4k
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 $T_d \to \mathbb{R}^3$? ...
- 13.6k
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 $M$. ...
- 68.9k
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) ...
- 10.4k
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\...
- 96.4k
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 ...
- 2,649
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 ...
CommunityBot
- 1
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
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 ...
- 6,962
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 ...
CommunityBot
- 1
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 ...
- 5,743
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 ...
- 96.4k
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 ...
- 28.2k