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

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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
22 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 ...
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1answer
28 views

Saccheri quadrilaterals: perpendicularity of midpoint segment, and comparative lengths of summit and base [on hold]

Help with a hyperbolic geometry problems (non-euclidean, Saccheri quadrilaterals)? Theorem. In Saccheri quadrilateral, the segment joining the midpoint of the summit and the base is perpendicular ...
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1answer
27 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 ...
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1answer
212 views

Fuchsian groups and topological isomorphism

I have a (finite) presentation of a group and I am wanting to prove that it is not Fuchsian. Because it is given by a presentation, a neat, algebraic description of Fuschian groups would be nice. This ...
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1answer
42 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 ...
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2answers
309 views

Riemann surface arising as a quotient of the upper half-plane.

Let $H$ be the upper half-plane $\{z \in \mathbb C \mid \Im(z) > 0\}$. For a fixed real $\lambda > 0$, let be the automorphism $$d_\lambda : H \to H, z \mapsto \lambda z .$$ Denote $\Gamma$ the ...
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1answer
56 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
185 views

Characterization of linearity in terms of metric

At least in Euclidean geometry and the upper half plane model of hyperbolic geometry, the statements '$y$ lies on the line segment determined by $x$ and $z$ ' and '$d(x,y)+d(y,z)=d(x,z) $' are ...
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1answer
22 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)} \, ...
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1answer
65 views

Fundamental solution of Poisson equation in the Hyperbolic Plane

If we consider the Poisson's equation $$ -\Delta u=f(x), \ \ \mbox{in} \ \ \mathbb{R}^n, $$ we can construct the fundamental solution $$ u(x)=\int_{\mathbb{R}^n}\Gamma(x-y)f(y)dy, $$ where $\Gamma$ is ...
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1answer
453 views

Aristotle's Axiom in Hyperbolic Geometry

I need to prove that Aristotle's Angle Unboundedness Axiom holds in hyperbolic geometry and I don't really know where to start. The problem says that we can take a segment parallel to one of the legs ...
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1answer
23 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 ...
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1answer
37 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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2answers
49 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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2answers
37 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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0answers
32 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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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
49 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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0answers
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
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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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1answer
197 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 ...
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2answers
18 views

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 ...
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1answer
25 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 ...
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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
23 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 ...
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1answer
63 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
33 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 ...
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1answer
36 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
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 ...
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1answer
91 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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1answer
22 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/ ...
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1answer
24 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 ...
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1answer
19 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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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 ...
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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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1answer
77 views

limit set of Kleinian groups with closed manifolds as quotient

I'm trying to convince myself that if $M\cong\mathbb{H}^3/G$ is a closed hyperbolic 3-manifold then the limit set $\Lambda(G)$ equals the whole Riemann sphere $S_\infty^2$. My idea of the proof goes ...
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0answers
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How to parameterize these pretty hyperbolic (Amsler) surfaces?

I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature. To this end, I attempted to model the ...
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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
37 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 ...
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0answers
33 views

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
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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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2answers
229 views

Poincare disk and Poincare half plane

My book claims that the Möbius transforms are isometries of the Poincaré half plane model. Thus, the metric is preserved under these maps. But I know that the Poincaré disk can be derived from ...
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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
26 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 ...
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1answer
34 views

Conics and Loci Question (Hyperbolae and Circles)

A circle has the equation $x^2 + y^2 = r^2$. Tangents are drawn from a point $P(x_1,y_1)$ to the circle and these touch the circle at points $A$ and $B$. If the position of $P$ can vary and the locus ...
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1answer
53 views

Parallelism preservation of hyperbolic rigid motions on the half plane model

I need to proof (Under Hilbert axiomatization) that hyperbolic rigid motions, with respect to the metric $ d:\mathbb{H}^2 \times\mathbb{H}^2\rightarrow\mathbb{R}: d(A,B) =\left| \log \left( ...
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4answers
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Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
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1answer
19 views

Parametrizaction of a Hyperboloid

I do not understand why when you revolve a hyperbola around a circle the respective parameters (cosh (v) and cos (u)) are multiplied by each other to get the parametric form of the hyperboloid. I ...