Questions tagged [hyperbolic-geometry]
The hyperbolic-geometry tag has no usage guidance.
885 questions
4
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Compact open topology on the space of geodesics
I'm new in the field, so I'm sorry in advance if my question is too naive.
Let's consider $S$ a surface of genus $g\ge 2$ with an hyperbolic metric $g$. Let's call $\mathcal{S}(S)$ the set of closed ...
4
votes
1
answer
256
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Action of the isometry group of the hyperbolic 5-space
We can think hyperbolic 5-space as, $$\mathcal{H}^5=SO^+_{5,1}(\mathbb{R})/SO_5(\mathbb{R})=SL_2(\mathbb{H})/Sp^*_2(\mathbb{H}),$$$\mathbb{H}$ is real quaternion algebra. By Iwasawa Decomposition the ...
- 1,692
4
votes
1
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228
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What's the height of the capped hyperbolic pants?
Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_1,l_2,l_3$. Cap off boundary 1 with a conformal disk. The result is a conformal cylinder, which has a unique flat ...
- 2,513
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548
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Pleated surfaces do not curl up too much
Hi!
Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is ...
- 213
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1
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613
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A regularity question on the Beltrami equation $ f_\bar{z} =\mu . f_z$ on $D$
Hello,
This question is related to Chapter V, lemma 3 on page 54 of Lars Ahlfors' 'Lectures on Quasiconformal mappings' which states :
If $\mu:\mathbb{C}\to \mathbb{D} \in W^{1,p}(\mathbb{C}), p ...
- 1,471
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votes
0
answers
139
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Understanding a "straightforward" application of the method of stationary phase for proving a trace formula on compact hyperbolic surfaces
I'm reading the paper 'Uniform distribution of eigenfunctions on compact hyperbolic surfaces' by Steven Zelditch (Duke Mathematical Journal, 55, pp. 919-941 (1987), MR916129, Zbl 0643.58029) and am ...
- 383
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0
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112
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MGFs of sum of (Rademacher) independent variables and (hyperbolic/spherical) Pythagorean theorem
Consider a set of iid random variables $X_1, X_2, \ldots$ (distribution to-be-specified later). For real numbers $a_1, a_2, \ldots$ (with $\sum_{k} a_k^2 < \infty$) define
$$X = a_1 X_1 + a_2 X_2 +...
- 637
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0
answers
77
views
Existence of finite 3-dimensional hyperbolic balanced geometry
Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
- 501
4
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0
answers
101
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Is there a 5-cell-600-cell honeycomb?
Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
- 2,773
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433
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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
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0
answers
172
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Survey or good reference of taut foliations
I am interested in the topology of foliations.
In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows.
I guess that
A. Candel and L. Conlon, Foliations I (...
- 73
4
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answers
106
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Geodesic foliations of open manifolds foliated by hyperbolic spaces
It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...
- 4,729
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172
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Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!
Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...
- 5,349
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88
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Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds
Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
When $M$...
- 481
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470
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Cluster algebras of type A and X
I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces.
Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
- 1,435
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0
answers
1k
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Definition of quasi-geodesics
I was going through the books Géométrie et théorie des groupes by Michel Coornaert, Thomas Delzant, Athanase Papadopoulos and Metric Spaces of Non-Positive Curvature by Martin R. Bridson, André ...
- 43
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173
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Random walks on the Poincaré disk
Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \...
- 401
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88
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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
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129
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What does hyperbolicity of curves buy us in the arithmetic context?
This is going to be a fairly vague question but hopefully it will have concrete answers:
There is this recurrent phenomenon in the arithmetic of curves (possibly stacky,affine) where there is a "...
- 7,746
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127
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Area lower bound given a mean curvature upper bound?
If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
- 41
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318
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Bers' simultaneous uniformization
I have been trying to understand Bers' famous paper "Simultaneous Uniformization". Regarding this paper I have a few questions. Any kind of help will be appreciated.
Let $S$ and $S^{'}$ be two ...
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206
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Explanation for phenomenon in hyperbolic geometry
By examining numerous examples I have become quite convinced that the following statement is true.
Let $x$ and $y$ be distinct points in the hyperbolic plane $\mathbb{H}$. Let $\gamma$ be the ...
- 1,769
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103
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Quadrics in $\mathbb{H}^3$
Consider a hyperbolic space $\mathbb{H}^3$ in the Beltrami-Klein model:
$$\mathbb{H}^3=\{(x,y,z|x^2+y^2+z^2\leq 1\}\subset \mathbb{R}^3.$$
Let $Q$ be a quadric in $\mathbb{R}^3.$
Question: What is a ...
- 4,316
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269
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Hyperbolic Intercept (Thales) Theorem
Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry?
Euclidean Intercept Theorem:
Let S,A,B,C,D be 5 points, such that SA, SC, ...
- 345
4
votes
1
answer
249
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Trace field of a hyperbolic $3$-manifold with a totally geodesic subsurface
Let $X$
be a complete finite-volume orientable hyperbolic $3$-manifold,
and let $\Gamma$
be a Kleinian representation of $\pi_1(X)$.
Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big)...
- 2,436
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70
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Given a CAT(0) space can one construct a CAT(-1) space that has the other space as boundary?
Two easy questions:
Given a metric space $(Z,d_Z)$, let $X = \text{Con}_h(Z) = Z \times ]0,\infty[$
and for $(z,i), (z',i') \in \text{Con}_h(Z)$ define a metric on $X$ as follows:
$d((z,i), (z',i')) =...
- 447
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206
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Exponential contraction for the projection on horospheres
A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a ...
- 2,090
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2
answers
258
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How should we define $\mathrm{PSL}_2$ of a Clifford group?
UPDATE - Feb. 9, 2017: The original title of this post was
"The $\text{isometry}^+$ group of hyperbolic $n$-space as $\mathrm{PSL}_2$ of a Clifford group."
The original question, which appears below,
...
- 2,436
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128
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Is there a formula for the A-model partition function in terms of hyperbolic structure?
The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant?
I'm not sure if there is a way to normalize these since the usual perscription $Z(S^2) ...
- 607
4
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0
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97
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Bi-Lipschitz classification of germs of conformal metrics at a singularity
First let me introduce some definitions.
By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in $...
- 1,804
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125
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Origin of spectral theory on infinite-area hyperbolic surfaces
The study of spectral theory of finite-area hyperbolic surfaces is intimately related to number theory, in particular by the importance of Maass cusp forms. The counting of resonances is of importance,...
4
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302
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Haken manifolds and characterising sutured manifold hierarchies
In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--...
- 367
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154
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reference request: “p-adic” presentation of surfaces
On several occasions I heart about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
- 51
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670
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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
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2
answers
641
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Computing hypergeometric function at 1
I'm looking to compute
$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr
1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$
for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...
- 337
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votes
3
answers
1k
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Poincaré metric on hyperbolic plane
As is well known, we can put a metric on the upper half plane $\mathbb{R}^+ \times \mathbb{R}$
by setting
$$
d\left((x,t);(x',t')\right):=\log\left(\frac{1 + \delta}{1 - \delta}\right)^{1/2},
$$
where
...
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Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface
I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
user147962
3
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1
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Hyperbolic 3 manifold with trivial deformation of flat conformal structures
Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?
- 171
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344
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Hyperbolic volume of hyperbolic knots
Let $G$ be a torsionfree kleinian group. Is there necessary an sufficient conditions on $G$ to be a knot group ?
It seems that there is some necessary conditions:
$H_{1}(BG) = \mathbb{Z}$
$H_{2}(BG) ...
- 223
3
votes
2
answers
950
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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
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submanifold of a hyperbolic manifold
Let $(M,g)$ be a compact orientable hyperbolic manifold with dimension at least $4$. Are there any topological conditions on $M$ which guarantee the existence of a hyperbolic surface in $M$? i.e. a $2$...
- 6,995
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1
answer
347
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Homotopy Groups of de Sitter and Anti-de Sitter?
Given $n$-dimensional de Sitter or Anti-de Sitter space, $dS_n$ or $AdS_n$, what are the homotopy groups $\pi_m(dS_n)$ and $\pi_m(AdS_n)$ and how does one calculate such things?
- 31
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Hyperbolic Metric on a Riemann Surface
From uniformization theorem, it is known that every conformal class of metrics on a genus-$g$ Riemann surface with $n$ punctures such that $2g+n\ge 3$ contains a unique hyperbolic metric. The ...
- 989
3
votes
2
answers
1k
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Hyperbolic pair of pants.
Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential simple arc $\sigma$ ...
- 217
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1
answer
886
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The smallest positive eigenvalue and the length of the shortest geodesic
I'm confused about some things concerning lengths of geodesics on Riemann surfaces and positive eigenvalues of the Laplacian. Moreover, I'm also interested in the relation between these two.
Let $X$ ...
- 9,497
3
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3
answers
1k
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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
293
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harmonic functions on hyperbolic manifolds with finite volume is constant?
Consider a hyperbolic manifold $M=H^n/\Gamma$ with finite volume. Suppose that there exists a harmonic function $u$ defined on $M$. Then is $u$ a constant? If $M$ is compact, yes. So I want to know ...
3
votes
1
answer
245
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Example of hyperbolic 3-fold with no embedded incompressible subsurfaces
Kahn-Markovic show that every hyperbolic 3-fold contains
an immersed $\pi_1$ injective surface. Are there any known examples
of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective
...
- 494
3
votes
1
answer
257
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Connectedness of the thick part of a hyperbolic manifold?
In a solution to a recent post : Fundamental group of a thick part of hyperbolic manifold, Igor Belegradek makes this claim that the thick part of a hyperbolic manifold is connected. To me it seems ...
3
votes
1
answer
407
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Explicit formula for embedding Cayley graph of free group into hyperbolic space
The problem is to embed Cayley graph of free group with $n\geq2$ generators (the same as Bethe lattice with coordination number $2n$) into any model of $\mathbb{H}^2$ (we have no model preference, the ...
