Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.

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1answer
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Is every hyperbolic isometry the restriction of an orthochronous Lorentz transformation?

I know that every isometry of the sphere $\Bbb S^2$ is the restriction of some $A \in {\rm O}(3,\Bbb R)$: namely, if $A_0:\Bbb S^2\to \Bbb S^2$ is an isometry, then $A_0 = A\big|_{\Bbb S^2}$ where ...
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2 views

Metric relations in lambert quadrilateral

I already found the relations in a rectangle triangle (6 formulas for the sides) and for a general ordinary triangle (sine and cosine hyperbolic laws). But now I'm trying to find them for a triangle ...
3
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0answers
61 views

How low will a given string hang? [on hold]

If I have a piece of string that is n meters long, attached at two points m meters apart, how low will the string hang? The two ...
2
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0answers
27 views

Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
0
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1answer
40 views

Area of circle in terms of Gaussian curvature

I am asking about a formula in section 2 of these notes. Let $\rho|dz|$ be a conformal metric on $U\subset\mathbf C$. Then the Gaussian curvature of $\rho|dz|$ at $z\in U$ is defined as ...
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0answers
15 views

Are Fuchsian groups without elliptic and parabolic elements at most countable? [duplicate]

Let $G \subset PSL(2, \Bbb R)$ be a discrete subgroup without elliptic or parabolic elements. Does it follow that it is at most countable? Subgroups as above have the property that the quotients of ...
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29 views

Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...
2
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1answer
29 views

Name of the modular group

I've been studying the hyperbolic plane and the action of the group $PSL(2,\mathbb{R})$ on it. I found that the modular group $PSL(2,\mathbb{Z})$ is a discrete subgroup of $PSL(2,\mathbb{R})$ so it's ...
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19 views

Studying the hyperboloid model, what is represented by the conic sections?

I am trying to get my head around the hyperboloid model of hyperboloic geometry https://en.wikipedia.org/wiki/Hyperboloid_model (article is much to technical please improve) And was thinking the ...
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18 views

Ergodic measure for action of $SO_2$ on lattice

Let $X:= \Gamma/PSL_2(\Bbb R)$ and for each $x \in X$ define $\phi_x(g):= xg^{-1}$ for $g \in SO_2$. Then the induced measure $(\phi_x)_*m_{SO_2}$ is ergodic for the $SO_2$ action and is a factor of ...
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12 views

Hyperbolic half-planes are geodesically-convex

I'm trying to understand the concept of Dirichlet domains associated to the action of a Fuchsian group $G$ on $\Bbb H$ (the upper half-plane of $\Bbb R^2$ endowed with its usual hyperbolic metric). ...
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27 views

Proof of $\delta$-Hyperbolicity of $\mathbb H^n$ just with the hyperboloid model?

Do you know any proof of the fact that $\mathbb H^n$ is Rips-hyperbolic (i.e., geodesic triangles are $\delta$-slim for some $\delta$, also called "Gromov-hyperbolic" in some contexts), which makes no ...
3
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2answers
44 views

The triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$

This answer made me wonder if there is a geometrical way to prove that the triangle group $(\alpha,\alpha,\alpha)$ is a subgroup of the triangle group $(3,3,\alpha)$. In other words, how can we ...
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1answer
16 views

Connected components of a subset of E

Let $E$ be a real vector space of dimension n+1 with a symmetric bilinear form B of signature (n,1). Let $H=\{x \in E : B(x,x) <0\}$. Somewhere I saw that it has two connected components. Can ...
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1answer
23 views

Geometrical interpretation of the conjugacy of triangle groups.

Let $\triangle$ and $\triangle'$ be two hyperbolic triangles of respective angles $\alpha,\beta,\gamma$ and $\alpha',\beta',\gamma'$. Let us suppose that the triangle subgroups ...
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2answers
34 views

Triangle group $(\beta,\beta,\gamma)$ is a subgroup of the triangle group $(2,\beta, 2\gamma)$.

Let $1/\alpha+1/\beta+1/\gamma<1$, and let us consider the triangle group $(\alpha,\beta,\gamma)$, i.e. the subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ induced by the hyperbolic triangle ...
0
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1answer
33 views

Show every Mobius transformation $T(z)=\frac{\alpha z+ \beta}{\bar \beta z+ \bar \alpha}$ acts as an isometry of the hyperbolic disk

Consider the unit disk $\mathbb{D}=\{z: |Z| < 1\} \subset \mathbb{C}$ equipped with the hyperbolic metric $g$ induced by $1$ form $ds=\frac{|dz|}{(1-|z|^2)}$ I am trying to show that every Mobius ...
2
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1answer
25 views

Is a map that preserves the hyperbolic distance biholomorphic?

Let $\lVert z \rVert_w = \frac{|z|}{1 - |w|^2}$ be the hyperbolic distance in $\mathbb{D}$, and let the hyperbolic metric be $d(z, w) = \inf_\gamma \int_0^1 \lVert \gamma'(t) \rVert_{\gamma(t)} \, ...
5
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1answer
69 views

Wallpaper groups for the hyperbolic plane

I would be grateful if someone could direct me to a reference that classifies the equivalent of the wallpaper groups (and the frieze groups and the point groups, if possible) for the hyperbolic plane, ...
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1answer
46 views

How to graph in hyperbolic geometry?

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line ...
0
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1answer
45 views

hyperbolic confusion: Is an apeirogon even a (closed) polygon?

Via Tesselation of the upper half plane via Ford Circles I was introduced to Ford circles ( https://en.wikipedia.org/wiki/Ford_circle note the wikipedia article has been updated since that question) ...
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1answer
29 views

Product of two elliptic isometries with distincts centers

I'd like to know why is the product of two elliptic isometries of the hyperbolic upper plan (or of the unitary disk) with distincts fixed points is parabolic or hyperbolic? PS: I only need it for ...
3
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2answers
39 views

Question on ideal triangle and hyperbolic distance

I'm asking a question about a construction due to Thurston. Let's consider a hyperbolic triangle (I'm considering the Poincarè disc model of the hyperbolic plane) and from each one of the three ...
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2answers
55 views

Tesselation of the upper half plane via Ford Circles

I have a question about the tesselation of the upper half plane via Ford Circles. Wikipedia says By interpreting the upper half of the complex plane as a model of the hyperbolic plane (the ...
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1answer
43 views

Find the curvature of the metric $ds=\frac{|dz|}{(1+|z|^2)}$ on $\mathbb{C}$

The curvature of the metric $g$ is defined as $$k(z)=-\bigg(\frac{2}{\alpha(z)}\bigg)^2 \partial \bar\partial log \alpha(z)$$ where $\alpha$ is positive and real valued. Also ...
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0answers
30 views

Does anything obstruct Mostow-Prasad rigidity for orbifolds?

If we phrase the Mostow-Prasad rigidity theorem algebraically, it goes like this (let $\mathcal{H}^n$ be a model for hyperbolic $n$-space). For $n>2$: if ...
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2answers
53 views

hyperbolic spaces and fractals

Is there a relation between hyperbolic spaces and fractals? In group theory, if we take the Cayley graph of a free group on two generators, we get a fractal quaternary tree, which I'd like to think as ...
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10 views

What is the geometric center and what is the other point?

In Euclidean geometry it is simple: In a triangle $\triangle ABC$ there is a single point $H_a$ on $BC$ such that the triangles $\triangle ABH_a$ and $\triangle ACH_a$ have the same area. the ...
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2answers
18 views

Finding the equation of a hyperbola given the vertices and foci.

A hyperbola has the vertices $(0,0)$ and $(0,-16)$ and the foci $(0,2)$ and $(0,-18)$. Find the equation with the given information.
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Triangle Inequality for Hyperbolic Metric (Logarithm of Cross-Ratio)

I need clarification in one step of this answer to my previous question. I was re-reading it, and it isn't clear to my why we can make the assumption of $ \Im(p)< \Im(q)< \Im(r) $. (Immaginary ...
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2answers
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Identifying the conic given some conditions.

So I have to identify the conic which represents the centre of the circle which touches another circle externally, and also touches the x axis. Here's a link to the exact question with the equation ...
8
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1answer
200 views

Lines in upper half-space

I'm teaching a tour-of-classical-geometry class this semester, and we are soon to introduce hyperbolic geometry. I am very inexpert in this subject, and I have a question about a compatibility of a ...
1
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1answer
28 views

saccheri quadrilateral - how does base=summit violate hyperbolic parallel axiom?

I drew diagonals across the quadrilateral and was able to prove that the summit angles are right angles by SSS and CPCTC. Therefore the two congruent triangles creats a quadrilateral with an angle sum ...
0
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1answer
13 views

Given two hermitian matrices of signature (2,1) there exists a Cayley transform between them?

Given a matrix $A\in M_{k\times l}(\mathbb{C})$ we define the hermitian transpose of $A$ as the matrix $A^*=\overline{A}^t\in M_{l\times k}(\mathbb{C})$. We say a matrix $H\in M_k(\mathbb{C})$ is ...
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1answer
25 views

Distance in Poincaré disk from origin to a point given

Let $C$ circle $x^2+y^2=1$ find the distance (Poincaré disk) from $O=(0,0)$ to $(x,y)$ The distance in Poincaré is $d=ln(AB,PQ)$ where AB are a segment of the curve and P and Q are points in the ...
2
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1answer
34 views

Formula for the midpoint in the hyperbolic geometry

I have two questions. First, is there a relatively simple formula for the midpoint of two points $a_1$ and $a_2$ in the disk with respect to the hyperbolic geometry? That is, the point on the ...
2
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1answer
37 views

How to visualize the region $\mathbb{H}/\Gamma_0(4)$ and its cusps?

In number theory we learn that $\theta(z) = \sum q^{n^2}$ is a modular form with respect to $\Gamma = \Gamma_0(4)$. This boils down to two properties: $\theta(z)= \theta(z+1)$ this shift symmetry ...
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1answer
69 views

Proof that three parallel lines don't be cutted by a transversal in Klein model

How do you prove that three parallel lines don't be cutted by a transversal? By definition parallel are Chords that meet on the boundary circle are limiting parallel lines. Then I built three ...
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1answer
15 views

hyperbolic trigonometry when one angle =0

The hyperbolic trigonometry functions don't really help when you have one angle =0 (the remaining lenght of side $AB$ becomes ${\infty}-{\infty}$ ) Given a triangle $\triangle AB \Omega$ with ...
0
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1answer
23 views

quasi-geodesics in hyperbolic space

I've stumbled across a proof of geodesic stability in hyperbolic space, located in the following blog post: https://lamington.wordpress.com/2010/05/19/hyperbolic-geometry-notes-5-mostow-rigidity/ ...
2
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1answer
26 views

Horocycle transformation in the Poincare half plane model

I was puzzeling with how to find an easy formula to calculate the length of a horocycle in the Poincare half plane model Then I had the brainwave that I can just use a transformation and then find ...
1
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1answer
21 views

About the congruence relation on Poincaré Half-Plane model

I've been studying Hyperbolic Geometry under Hilbert Axiomatization on the Poincaré Half-Plane model. The congruence relation of segments is defined as $AB \equiv CD \Leftrightarrow \exists L \in ...
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18 views

Hyperbolic isometries preserve hyperbolic paralelism

In the Poicaré half-plane model, under Cayley-Klein metric, i. e., $ d:\mathbb{H}^2\rightarrow\mathbb{R}, d(A,B) = \big{|} log ...
2
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1answer
39 views

Proof of an identity that relates hyperbolic trigonometric function to an expression with euclidean trigonometric functions.

Given a line $r$ and a (superior) semicircle perpendicular to $r$, and an arc $[AB]$ in the semicircle, I need to prove that $$ \sinh(m(AB)) = ...
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2answers
37 views

Length of parametrized path

Can someone guide me through how to solve this problem? Let $P = (0,1)$ and $Q = (1,1)$, and let $\gamma$ be the following parametrized path in $\mathbb H^2$ from $P$ to $Q$: $\gamma(t) = (t,1)$. ...
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1answer
41 views

Going from Metric to Distance Function in the Poincaré Half Plane

Let the Poincaré Half Plane be the set $\{(x, y) \in \mathbb{R}^2 : y > 0\}$. It is a known result that the the metric $ds^2 = \frac{dx^2 + dy^2}{y^2}$ yields a distance function $f$ such that ...
2
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1answer
100 views

Hyperbolic circles are euclidean circles in the Poincaré Half Plane Model

Consider the metric space $(\mathbb{H}²,d_{\mathbb{H}^2})$, where $d_{\mathbb{H}^2}$ is the hyperbolic Cayley Klein metric, i.e., $ d_{\mathbb{H}^2}(A,B) = |log ((AA_{\infty}. BB_{\infty}) / ...
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0answers
23 views

Formula for length of diagonal in a Lambert quadrilateral

Given a Lambert quadrilateral $AOBF$ where the angles $ \angle FAO , \angle AOB , \angle OBF $ are right, and $F$ is opposite $O , \angle AFB$ is the acute angle , and the Gaussian curvature = -1 (so ...
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27 views

The action of an S-arithmetic group on the hyerbolic plane

I have a really quick question. I am interested in $G=SL_2(\mathbb{Z}[1/p_1,...,1/p_n])$, where $p_1,..., p_n$ are prime numbers. Since $G$ is a subgroup of $SL_2(\mathbb{R})$, it acts in the ...
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1answer
28 views

How to build a hexagon according to Poincaré model?

Given a side, I know how to build a hexagon in the euclidean geometry. How can i build it in the hyperbolic geometry according to the Poincaré model? By translating every step using hyperbolic circle ...