The hyperbolic-geometry tag has no usage guidance.

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### Gauss Bonnet theorem calculation for pseudosphere

In an attempt to verify Gauss-Bonnet theorem for a Beltrami pseudosphere I calculated a simple case of the Riemann sphere. Taking curved radius $a$ for geodesic polar co-ordinate from the smooth ...

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**1**answer

121 views

### Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces

I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using ...

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183 views

### Why are the medians of a triangle concurrent? In absolute geometry

This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...

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**1**answer

80 views

### Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?

Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
What if in Question 1 we restrict the cone angles to be $...

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101 views

### Simple Closed Hyperbolic Geodesics on Punctured Spheres

Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...

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305 views

### Geodesic current supported on a pencil?

Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...

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**1**answer

62 views

### Intersecting geodesics on a surface from non-intersecting geodesics

Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?

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70 views

### delta hyperbolicity of a product of poincare balls? [closed]

I know how to compute the $\delta$-hyperbolicity of a $2D$ poincaré disk of radius 1, but I was wondering how to generalize such computations to:
Higher dimensions
Poincaré disk with radius $r$ and ...

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**1**answer

144 views

### Cusps in hyperbolic manifolds and fundamental group

I am reading the book "The Arithmetic of Hyperbolic Three Manifolds" by Maclachlan and Reid and I am having some problems in understanding something about cusps.
The definition they give of a cusp is ...

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**1**answer

69 views

### Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$?

I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me.
The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ ...

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34 views

### First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...

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72 views

### 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 ...

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785 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 ...

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**1**answer

184 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 ...

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44 views

### Characteristics of Poincare's ball model (of hyperbolic spaces)

Are there any references (papers, books, etc.) that have ready-calculated equations for geometric quantities of the hyperbolic space in terms of the coordinates of Poincare's ball models?
By ...

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259 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 $G$ ...

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113 views

### Expected distance in hyperbolic space

In a hyperbolic space, $r$ and $\theta$ can represent a point in a polar coordinate system. If we suppose $\theta_1\sim \operatorname{Uniform}(t_1,t_2)$, $\theta_2\sim \operatorname{Uniform}(t_3,t_4)$,...

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171 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 $(\mathcal{P} \to M, \omega)$ ...

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83 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 ...

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**1**answer

174 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 ...

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**1**answer

115 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 ...

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**1**answer

840 views

### Four circles on the sphere

Consider 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 enumerate/...

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350 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 ...

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196 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(...

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**1**answer

49 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 ...

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198 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 ...

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150 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?

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204 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.
...

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103 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 ...

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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 ...

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156 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 ...

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88 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 ...

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221 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 $ ...

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318 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 ...

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85 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 ...

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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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41 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 ...

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**1**answer

95 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.
...

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108 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 ...

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138 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 ...

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90 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 ...

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183 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 ...

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242 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!

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651 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 ...

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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 $X$. ...

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230 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. ...

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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 ...

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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'...

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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)}{...

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126 views

### Lemma involving sinh and cosh [closed]

I need a small lemma for a paper I am working on and I claim:
Lemma 1: For every $C >0$ there exist a $c_0 >0$ such that for all $a,b,c,d > c_0$:
$$\sinh(a)\cosh(b) \geq \sinh(c)\cosh(d)$$
...