Questions tagged [hyperbolic-geometry]

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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 ...
user avatar
24 votes
4 answers
1k views

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 ...
alvarezpaiva's user avatar
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3 votes
1 answer
344 views

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 ...
user avatar
7 votes
4 answers
415 views

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 $G$ ...
Johannes Sunstein's user avatar
3 votes
2 answers
443 views

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 $(\mathcal{P} \to M, \omega)$ ...
ಠ_ಠ's user avatar
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11 votes
0 answers
255 views

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 ...
giulio bullsaver's user avatar
5 votes
1 answer
514 views

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 ...
user avatar
3 votes
1 answer
155 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 ...
Xin Nie's user avatar
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40 votes
1 answer
1k views

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 ...
Alexandre Eremenko's user avatar
6 votes
2 answers
570 views

$S^3 \setminus S^1$ 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 ...
Mykola Pochekai's user avatar
6 votes
1 answer
499 views

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(...
john mangual's user avatar
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2 votes
1 answer
106 views

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 ...
Florian R's user avatar
  • 215
5 votes
1 answer
279 views

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 ...
David Sun's user avatar
  • 309
5 votes
1 answer
219 views

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?
Vanderson Lima's user avatar
3 votes
0 answers
323 views

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. ...
geometer13's user avatar
4 votes
1 answer
878 views

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 ...
davidv1992's user avatar
1 vote
0 answers
311 views

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 ...
user123723's user avatar
4 votes
1 answer
293 views

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 ...
Vanderson Lima's user avatar
0 votes
1 answer
262 views

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 ...
Ferran V.'s user avatar
  • 627
9 votes
1 answer
360 views

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 $ ...
Ali Taghavi's user avatar
10 votes
1 answer
680 views

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 ...
M. Dus's user avatar
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2 votes
0 answers
124 views

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 ...
user2554's user avatar
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1 vote
0 answers
46 views

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 ...
giulio bullsaver's user avatar
1 vote
0 answers
60 views

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 ...
giulio bullsaver's user avatar
1 vote
1 answer
332 views

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. ...
giulio bullsaver's user avatar
5 votes
1 answer
141 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 ...
aglearner's user avatar
  • 14k
2 votes
1 answer
265 views

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 ...
giulio bullsaver's user avatar
1 vote
0 answers
184 views

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 ...
giulio bullsaver's user avatar
3 votes
1 answer
549 views

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 ...
giulio bullsaver's user avatar
5 votes
1 answer
378 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!
Markiff's user avatar
  • 303
10 votes
2 answers
797 views

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 ...
Nick Salter's user avatar
  • 2,770
2 votes
0 answers
66 views

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 $X$. ...
Alex Gavrilov's user avatar
5 votes
2 answers
393 views

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. ...
Christopher King's user avatar
3 votes
0 answers
70 views

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 ...
user avatar
20 votes
2 answers
2k views

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'...
Christopher King's user avatar
7 votes
1 answer
459 views

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)}{...
M. Dus's user avatar
  • 1,900
3 votes
0 answers
186 views

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 ...
user avatar
7 votes
1 answer
451 views

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 ...
asldjk's user avatar
  • 318
0 votes
1 answer
122 views

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 $S^3$" (...
asldjk's user avatar
  • 318
3 votes
1 answer
399 views

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 ...
DHall's user avatar
  • 33
8 votes
0 answers
306 views

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 ...
Eric Peterson's user avatar
5 votes
2 answers
1k views

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 ...
user2554's user avatar
  • 1,869
0 votes
1 answer
121 views

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 ...
user117580's user avatar
3 votes
0 answers
230 views

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 ...
j0equ1nn's user avatar
  • 2,438
4 votes
1 answer
204 views

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)...
j0equ1nn's user avatar
  • 2,438
2 votes
1 answer
103 views

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 $\mathcal{I}^3$. ...
j0equ1nn's user avatar
  • 2,438
1 vote
0 answers
139 views

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 ...
Cusp's user avatar
  • 1,703
6 votes
3 answers
514 views

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. ...
Anubhav Mukherjee's user avatar
1 vote
1 answer
110 views

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 ...
gm01's user avatar
  • 325
2 votes
0 answers
85 views

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 ...
Yellow Pig's user avatar
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