Questions tagged [hyperbolic-geometry]
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861
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Density of closed orbits on hyperbolic surfaces
It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense.
My questions:
If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...
24
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4
answers
1k
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Immersions of the hyperbolic plane
Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was ...
3
votes
1
answer
344
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Question on a proof of density of periodic orbits
In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...
7
votes
4
answers
415
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Lattices of PU(n,1) with large abelianization
I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...
3
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2
answers
443
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Is the development map in Hyperbolic geometry related to development in Cartan geometry?
I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ ...
11
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0
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255
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Cluster algebra and Fenchel Nielsen coordinates
Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...
5
votes
1
answer
514
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How many simple closed geodesics in a given primitive homology class?
It is well-known that an essential closed curve on a hyperbolic surface (possibly with boundary) is homotopic to a unique closed geodesic. Moreover, if the curve under consideration is simple, then so ...
3
votes
1
answer
155
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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 ...
40
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1
answer
1k
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Four circles on the sphere
Consider generic configurations consisting of 4 distinct circles on the sphere.
Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to ...
6
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2
answers
570
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$S^3 \setminus S^1$ doesn't have hyperbolic structure
I need to prove that $M = S^3 \setminus S^1$ doesn't admit any metric of constantly negative sectional curvature s.t. $M$ is complete respect for this metric. I know that it is consequence of famous ...
6
votes
1
answer
499
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What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$?
I have been trying to learn about congruence groups. Here is an example:
\begin{eqnarray*} \Gamma\big(1+2i\big) &=& \text{SL}_2\big(\mathbb{Z}[i]\big)(1+2i) \\ \\
&=& \left\{ \left(...
2
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1
answer
106
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Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map
Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...
5
votes
1
answer
279
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Conformal boundary and cusp of figure-8 complement
As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...
5
votes
1
answer
219
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Intersection of $\pi_1$-injective surfaces
Let $M$ be a closed non-Haken hyperbolic $3$-manifold. Are there two, nonhomotopic, $\pi_1$-injective closed surfaces in $M$, and which do not intersect?
3
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0
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323
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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.
...
4
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1
answer
878
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Harmonic analysis on constant curvature hyperbolic spaces of arbitrary dimension
I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the ...
1
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0
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311
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Lifts of geodesics on surfaces onto the universal cover [closed]
self-intersecting geodesic on hyperbolic surface of genus 2
Given a self intersecting geodesic on a hyperbolic surface of genus 2 as in the picture, how can I understand precisely what the lift to ...
4
votes
1
answer
293
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Immersed incompressible surfaces in surface bundles
Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not ...
0
votes
1
answer
262
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Hyperbolic structures on infinite type surfaces
Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do ...
9
votes
1
answer
360
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Are two triangles with equal corresponding medians, congruent?
Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
Two triangles with equal corresponding medians are congruent.
More precisely: Assume that $\Delta ABC$ and $ ...
10
votes
1
answer
680
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Parabolic subgroups of relatively hyperbolic and CAT(0) groups
Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...
2
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0
answers
124
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Explanation of an unexplored note of Gauss on hyperbolic volume
My question refers to Gauss's note "Cubirung Der Tetraeder", which can found at p. 228 in the section on the foundations of geometry in volume 8 of his collected works. In this note, Gauss wrote down ...
1
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0
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46
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Real section of moduli space of Riemann surfaces
In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset ...
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0
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60
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Annuli and pinched annuli vs circles and horocycles
Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed.
If we want to parametrise annuli with punctures at one boundary, give the punctures ...
1
vote
1
answer
332
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Parabolic elements of the Poincare' disk automorphism group as limit of elliptic ones
The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point.
...
5
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1
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141
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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 ...
2
votes
1
answer
265
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Build a Fuchsian group starting from punctures on a disk
Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$.
$\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the ...
1
vote
0
answers
184
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Cutting a circle from the hyperbolic plane
Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface ...
3
votes
1
answer
549
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What is a half cusp in hyperbolic geometry?
I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...
5
votes
1
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378
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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!
10
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2
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797
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Does Helly's theorem hold in the hyperbolic plane?
The Helly theorem in the Euclidean plane asserts that if $S_1, \dots, S_n$ are $n \ge 3$ convex subsets such that $S_i \cap S_j \cap S_k \ne \emptyset$ for all distinct triples $i,j,k$, then the total ...
2
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0
answers
66
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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 $X$. ...
5
votes
2
answers
393
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Smallest tile to *isohedrally* tessellate the hyperbolic plane
Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
3
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0
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70
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Does the orbital function divided by the volume of a ball decrease?
Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...
20
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2
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2k
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Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
7
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1
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459
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Rational stable translation length
Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...
3
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0
answers
186
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Ending lamination theorem
Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations ...
7
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1
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451
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Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?
I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.
Following this paper by Christian ...
0
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1
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122
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What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?
I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (...
3
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1
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399
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Classifying links with essential annuli in the complement as torus links
I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and ...
8
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0
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306
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Lines in upper half-space
A couple of years ago, I taught an undergraduate class introducing various aspects of classical geometry, learning the (beautiful!) subject as I went. I found one thing that really bothered me: the ...
5
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2
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How to derive from Gauss's results on the volume of hyperbolic orthoscheme tetrahedron the formula of Bolyai?
In his biography of Gauss, G. Waldo Dunnington describes the Gauss-Bolyai episode and their correspondence. In particular, he describes the contents of one letter from Gauss to Janos-Bolyai:
In the ...
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1
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121
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What’s the form of Gram matrix for right-angled hexagon
Informally, right-angled hyperbolic hexagon is a hyperbolic triangle with vertices outside infinity. I think there should be a Gram matrix for it, and what does it looks like?
(The Gram matrix here ...
3
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0
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230
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Pairs of non-isometric subsurfaces of a hyperbolic 3-manifold, with the same genus
When I say manifold below, I mean a complete orientable finite-volume hyperbolic $3$-manifold and when I say subsurface, I mean immersed closed totally geodesic subsurface.
Whenever a manifold has a ...
4
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1
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204
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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
votes
1
answer
103
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Find the fixed geodesic of an orientation-preserving isometry of the $3D$ hyperboloid model
Let $\mathcal{I}^3\subset\mathbb{R}^4$
be the standard hyperboloid model for hyperbolic $3$-space and consider the usual $\mathrm{SO}(3,1)$
action of $\mathrm{PSL}(2,\mathbb{C})$
on $\mathcal{I}^3$.
...
1
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0
answers
139
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Topological entropy and pseudo-Anosov dilatation for punctured surface
Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ punctures. Assume $2-2g-n<0$. Let $f$ be a pseudo-Anosov mapping class with dilatation $\lambda_f$. In the introduction (1st page) of the ...
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514
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Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
1
vote
1
answer
110
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About some lines on the universal covering of the punctured plane
Consider a finite set (of cardinality $\ge 2$) $S \subseteq \mathbb{C}$ and the holomorphic universal covering map $\pi: \ \mathbb{H} \rightarrow \mathbb{C} \setminus S$, where $\mathbb{H}$ denotes ...
2
votes
0
answers
85
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Hausdorff dimension of radial limit sets for divergence type subgroups
Let $X$ be a proper $CAT(-1)$ space.
Let $\Gamma<Isom(X)$ be a subgroup of divergence type.
Is it true that the Hausdorff dimension of the radial limit set of $\Gamma$ in $\partial X$ is equal to ...