# Questions tagged [hyperbolic-geometry]

The hyperbolic-geometry tag has no usage guidance.

607
questions

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### Maximally symmetric hyperbolic 3-manifolds with finite volume

In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...

**7**

votes

**1**answer

116 views

### Cutting up the Bring surface into six pairs of pants

The Bring sextic, with 120 automorphisms, is the numerically most symmetric compact Riemann surface of genus 4. To cut it up into six pairs of pants, we need to cut along nine disjoint geodesic loops....

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96 views

### Purely analytic proof of the Nielsen-Thurston classification theorem

I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory
and Applications to ...

**1**

vote

**0**answers

28 views

### Associativity property of the gyrobarycenter

I'm using Ungar's terminology and notations. In the open unit ball of $\mathbb{R}^n$, let $GB(A_1, \ldots, A_N; m_1, \ldots, m_N)$ be the gyrobarycenter of the points $(A_1, \ldots, A_N)$ with ...

**11**

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**4**answers

805 views

### Surfaces with non-constant negative curvature

Are there any nice models of surfaces with non-constant negative curvature, analogous to the Poincare disk for constant negative curvature. I have found lots of general results and theory but no nice ...

**2**

votes

**1**answer

56 views

### Coordinates for Laminations: geometric versus shear

Let $S$ be an orientable surface with a triangulation T.
A lamination $\ell$ is a simple closed curve on $S$, up to isotopy. We will assume that $\ell$ is drawn in such a way that it intersects the ...

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55 views

+150

### Bounds on convergence of two orbits in the limit set of a Schottky group

Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...

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votes

**1**answer

227 views

### Mostow Rigidity Theorem and reconstruction from fundamental group

The Mostow Rigidity Theorem is phrased in terms of a relationship between isometries and isomorphisms of fundamental groups, which raises an obvious question. Given the fundamental group of a complete ...

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22 views

### Explicit check of the invariance of the Weyl-Petersson form

Using Fenchel-Nielsen coordinates, the Weyl-Petersson metric can be written as
$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$
where $i$ is an index labelling the curves of a pants decomposition ...

**1**

vote

**1**answer

78 views

### Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?

Does there exist a real-valued function on the hyperbolic plane which has bounded hessian norm and unbounded gradient norm?
Specifically, consider the poincare half-plane model of the 2d hyperbolic ...

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**1**answer

67 views

### Is it possible to tessellate a torus minus a disk using hyperbolic right-angled pentagons?

I am trying to construct a compact hyperbolic surface tessellated with hyperbolic right-angled polygons with $n \ge 5 $ edges. I found quite easily a way to do it for $n$ even, but the odd case seems ...

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**2**answers

121 views

### When existence of loxodromic, WPD elements implies an action is acylindrical

Definitions
Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries.
Recall that an action of $G$ on $X$ is called acylindrical if the ...

**5**

votes

**1**answer

147 views

### The stabilizers of the canonical boundary action of hyperbolic groups

My question is that
Is every stabilizer of the canonical boundary action of a hyperbolic group on its Gromov boundary a finitely generated group?
I guess every stabilizer is a (finitely generated) ...

**8**

votes

**0**answers

244 views

### Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...

**2**

votes

**1**answer

201 views

### Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures

In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...

**5**

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**1**answer

246 views

### Hyperbolic manifolds with infinite cyclic fundamental group

It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...

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**1**answer

105 views

### Copies of $\mathbb{Z}\oplus \mathbb{F}_2$ in non-affine, irreducible Coxeter groups

Let $\left(W,S\right)$ be a non-affine, irreducible Coxeter system and assume that $W$ contains a copy of $\mathbb{Z}\oplus\mathbb{Z}$ (this is equivalent to $W$ being not word hyperbolic). Does this ...

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**1**answer

221 views

### Thickness and hierarchical hyperbolicity

Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here.
I've heard that it is open ...

**2**

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**0**answers

150 views

### Action on the upper half plane of the double coset of an irrational matrix in $\mathrm{SL}_2(\mathbb{R})$

Consider the action of $\mathrm{SL}_2(\mathbb{R})$ on the upper half plane $\mathbb{H}$ by Möbius transformations. Denote by $\Gamma$ the group $\mathrm{SL}_2(\mathbb{Z})$.
It is known that if $z \in ...

**4**

votes

**1**answer

154 views

### Visualizing hyperbolic metric of punctured sphere

Uniformization of the 3-punctured sphere generates a "pants" configuration with three legs narrowing down to cusps. This is supposed to have a metric of constant negative curvature, and I can see this ...

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**1**answer

187 views

### Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space

Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂𝑋 be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich https://...

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113 views

### Riemannian metric over moduli space of Riemann spheres with n punctures

In the paper `Tessellations of moduli spaces and the mosaic operad' by Devadoss (https://arxiv.org/pdf/math/9807010.pdf), on page 5-6, the author identifies hyperbolic planar tree space (or the ...

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44 views

### finite subgroups of discrete arithmetic groups

Let K be a totally real multi-quadratic fields and let $\mathcal{O}$ be its ring of integers. I would like to compute the orders of the finite subgroups of the discrete group $\mathrm{SL}_{2}(\mathcal{...

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157 views

### 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 ...

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vote

**1**answer

108 views

### Example of zero Lyapunov exponentes

Assume that $(T, A)$ is a linear cocycle such that $T:X\rightarrow X$ is a homemorphism on compact metric space $X$ and $A:X\rightarrow SL(2, \mathbb{R})$ is a continuous function.
We say that an ...

**5**

votes

**1**answer

251 views

### Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement

I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper https://arxiv.org/pdf/hep-th/9812206.pdf whose Eq. (9) mentions a theorem ...

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34 views

### Smoothness of boundary of $r$-neighborhood of convex core

The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. ...

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30 views

### Existence of eigenvalues and eigenvalues of infinite multiplicity in geometrically finite manifolds with infinite volume

In the paper, The geometry and spectra of hyperbolic manifolds https://link.springer.com/article/10.1007/BF02830802 by PETER D HISLOP, the author sketched a proof for the following theorem:
Let $\...

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**1**answer

151 views

### Equivalence of harmonic measures on hyperbolic groups

Consider a Gromov-hyperbolic group $\Gamma$ and let $\mu$ be a finitely supported probability measure on $\Gamma$. Assume that the support of $\mu$ generates $\Gamma$ as a semi-group, in other words, ...

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77 views

### Intersecting surface on pseudosphere

It is known what surface cuts a sphere resulting in intersections along great circle elliptic geodesics... this is any plane through its center.
What parametric surface cuts a pseudosphere resulting ...

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105 views

### 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 ...

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votes

**2**answers

229 views

### Area method in Lobachevskian geometry

There are many proofs in Euclidean geometry using the area method; for example, Ceva's theorem or the proof of Pythagorean theorem shown below.
Do you know such proofs in hyperbolic geometry?
I ...

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33 views

### Effect of plumbing a surface on the marked length spectrum

First I'll recall the plumbing procedure.
Let $M$ be a noded Riemann surface with nodes $p_1,\dots, p_n$. There is a family of pairwise disjoint neighbourhoods of each node $U_i$ that has coordinates ...

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55 views

### Arithmetic product and sum of limit sets of non-elementary Fuchsian group of second kind

Let $L \subset \mathbb{R}$ be a limit set of a Fuchsian group $\Gamma$. If $\Gamma$ is a non-elementary Fuchsian group of second kind, then $L$ is a Cantor set. For example: $\Gamma= \bigg\langle \...

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105 views

### 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 ...

**5**

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**1**answer

230 views

### Relations between boundaries of groups acting on hyperbolic spaces with WPD elements

Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...

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**4**answers

497 views

### Introductory textbook on geometry of hyperbolic space

I am looking for an introductory textbook to the geometry of the hyperbolic space $\mathbb{H}^n$. The book should include explicit description of geodesics and horospheres in various models (...

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190 views

### How the hyperbolic metric changes when we add a puncture?

Suppose we have a surface of a finite genus, without boundary with a finite number of punctures. This surface admits a unique hyperbolic metric of curvature $-1.$
If I add a puncture somewhere, the ...

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votes

**1**answer

59 views

### Horospherical distance in CAT($-1$) space

In $\mathbb{H}^n$, equipped with its hyperbolic metric of constant curvature $-1$, if we have two points $p,q$ on a common horosphere $\partial S$, then $$d_{\mathbb{H}}(p,q) = 2\sinh^{-1} (d_{\...

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194 views

### 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 ...

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votes

**1**answer

236 views

### On Thurston's triangulations of sphere

I have two questions from Thurston's paper [1].
In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...

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124 views

### Effect of the inverse exponential map on the curvature of a given curve

Suppose you have a curve $\alpha$ in a manifold $\mathcal{M}$. You are at a point $\alpha(t)$ of that curve. The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}...

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92 views

### 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 ...

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**1**answer

158 views

### Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups

Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...

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66 views

### Bound on the distance from points to the boundary of a hyperbolic surface

Fix $\epsilon\in\mathbb{R}_{>0}$, $\Sigma$ a surface with boundary and let $\mathcal{T}_{\Sigma}(L_{1},...,L_{n})$ denote the Teichmüller space of hyperbolic structures of $\Sigma$ with geodesic ...

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votes

**3**answers

151 views

### Teichmuller space for surface with cone points

Working on my current research problem, Teichmuller spaces for surfaces with cone points have come into play. It's fairly easy to formulate some of the definitions, a few basic results, etc. To be ...

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56 views

### Pseudoisometry in Mostow's rigidity theorem for non-compact manifolds

The first step of all proofs of Mostow's Rigidity theorem for closed manifolds is to show that given a homotopy equivalence $M\rightarrow N$ between two closed hyperbolic manifolds, having the same ...

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124 views

### What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?

What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?
Some sources seem to imply that they are the same, i.e. having at least the same symmetry group $\mathrm{SL}(2,...

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59 views

### Converse to Wolpert's Lemma

Recall Wolpert's lemma:
Let X,Y be hyperbolic surfaces and $f:X\to Y$ a $K$-quasiconformal homeomorphism. For any homotopy class of curves $c$ let $\ell(c)$ denote the length of the geodesic in the ...

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130 views

### Finitely generated nilpotent groups as cusp groups

I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...