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

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3
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1answer
42 views

Two transformation groups of the hyperbolic plane are isomorphic?

I'm aware that $PGL_2(\mathbb{R})\simeq GL_2(\mathbb{R})/\mathbb{R}^\times$ is isomorphic to the full isometry group of $H^2$, the hyperbolic plane. I've just been told that $SO(2,1)$, the indefinite ...
2
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1answer
44 views

Hyperbolic Geometry: Question about the Transitivity of Möbius transformations

I was confronted with this exercise in the book Hyperbolic Geometry by Anderson which states: In each case, find $m \in Möb(\mathbb{H})$ such that the property holds, or prove that no such $m$ ...
2
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1answer
27 views

Question about hyperbolic space-forms

I have a map between hyperbolic space-forms $\varphi:B^n/\Gamma \longrightarrow B^n/H$ (where $\Gamma, H$ are discrete groups of isometries that act freely), and a lift to a map $\tilde{\varphi}:B^n ...
2
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1answer
61 views

Broken geodesics in the hyperbolic plane and bending angles

Let $\gamma$ be an infinite broken geodesic in the hyperbolic plane, that is a curve formed by consecutive geodesic segments. Assume also that each of these segments is longer than a certain positive ...
2
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1answer
143 views

Seifert manifolds and Fuchsian group

A Fuchsian group is a discrete subgroup of $PSL(2, \mathbb{R})$. Let $M$ be a Seifert manifold (maybe with boundary) and $t \in \pi_1(M)$ the class of a regular Seifert fiber. Hempel claims in his ...
2
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1answer
62 views

Right angles in hyperbolic pool

(This uses a bit of physics) So I learned today the following fact from physics: Imagine you have two pool balls of the same mass. You hit the first one, and it collides into the second. Then their ...
1
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1answer
6 views

Finding the euclidean centers of the geodesics AB, AC, and BC

I am trying to learn about finding the angles in hyperbolic geometry and I am trying to understand this example given in Stahl's Introduction to topology and geometry. You can notice that there is a ...
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1answer
40 views

Extending the metric of a hyperbolic surface with boundary to its double

Let $M$ be a hyperbolic surface with totally geodesic boundary. Taking the double $DM$ of $M$, it is easy to see using Euler characteristic that $DM$ is itself a hyperbolic surface (without boundary). ...
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1answer
51 views

Is there any textbook about computing the automorphism group of the triangle group?

For example computing the automorphism group of the 2 genus surface made by triangles (12,2,3) in the hyperbolic plane. In addition,if you know the trick of the computing the automorphism groups like ...
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1answer
20 views

Canonical map from fundamental group to Fuchsian group?

Suppose we have a Riemann Surface $S$ of constant negative curvature $-1$. What is the canonical map from the fundamental group $\pi_1(S)$ to the discrete subgroup $\Delta \subset PSL_2(\mathbb{R})$ ...
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1answer
80 views

(compact, non-empty boundary )Surface Geodesics on Hyperbolic Geometry

I have a basic knowledge of hyperbolic geometry . I am trying to understand the meaning of "a compact surface S with non-empty boundary(which is neither a disk nor an annulus )with a complete ...
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1answer
68 views

The distance in Lobachevski (Hyperbolic) space

I need to find the distance from the point provided in the hyperboloid model with a vector $x$ where $\langle x,x\rangle=-1$ to the hyperplane $H_e$ with a normal vector $e$, where $\langle ...
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1answer
35 views

Definition of complex hyperbolic geometry

I am trying to read about complex hyperbolic geometry.But I couldnot find a basic definition for it. Is it just the special case of hyperbolic geometry where we work with complex numbers in the model. ...
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1answer
135 views

hyperbolic geometry (and circle ) construction problem

Was thinking about hyperbolic geometry, the Poincare Disk Model and Sweikarts constant and combined them all in a construction puzzle that I was unable to solve. My construction puzzle: Given: A ...
1
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1answer
51 views

hyperbolic equilateral triangle : $\cosh \left(\frac{1}{2} a\right) \sin \left(\frac {1}{2} \alpha\right) = \frac{1}{2}$

I met this problem in Ratcliffe's Foundations of Hyperbolic Manifolds. Please help me prove this. In an equilateral triangle with side length $a$ and angle $\alpha$, $$\cosh \left(\frac{1}{2} ...
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1answer
275 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 ...
1
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1answer
42 views

L is a family of hyp-lines passing through a pt. Not sure how this implies rr'=(c−Re(p))c'

Lemma 1. Let $p \in \mathbb{H}$, and assume $l$ is a family of hyp-lines passing through $p$ such that $l$ is of the form $l = \{c +re^{i\theta} | 0 < θ < π\}$. For simplicity, assume the ...
0
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1answer
21 views

Find the 3 angles of the hyperbolic triangle

A(0,5) B(0,2) C(4,2) In Euclidean geometry the three points given are the vertices of a right-angled triangle. Find the three angles of the hyperbolic triangle with vertices A,B,C. Find the ...
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1answer
30 views

calculating Hyperbolic distance

To calculate the hyperbolic distance we use the formula $$\left|\frac{w-z}{1-\bar wz}\right|$$ I want to apply this to the following pair of points: \begin{align*} w&=\frac{-1}{\sqrt{3}}\space ...
0
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1answer
40 views

A casual definition of Hyperbolic Space

I'm writing an article about a lecture that mentioned hyperbolic space. I wondered if anyone had a friendly way of describing it to the general public. (I will rewrite any definitions in my own ...
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1answer
40 views

Triangle Identity leads to another Euclidean parallel.

Referring to TriangleIdentity by 伍柒貳 a while ago, considering $\bigtriangleup$ ABC, it is proved that: $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C (1*) $$ I want to take angle $A = ...
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1answer
53 views

Discrete faithful representation in $PSL(2,\mathbb R)$ and horocycles in hyperbolic space

Let $S$ be a closed oriented surface of genus $g>1$. Is the following true ? Let $\alpha,\beta\in \pi_1(S)\backslash \{1\}$ and $\rho:\pi_1(S)\rightarrow PSL(2,\mathbb R)$ be a discrete ...
0
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1answer
42 views

Semiregular tilings of the hyperbolic plane

Consider the irregular quadrilateral tiling of the Euclidean plane depicted by the log-log coordinate grid: I'm wondering if in the Hyperbolic plane exist some analog of this kind of tiling where ...
0
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1answer
37 views

measuring curvature

Suppose you are transported to an 2 dimensional hyperbolic world, ( a plane (2 dinensional) manifold with a constant negative curvature ) the only geometrical tools you have are a ruler, a pencil, ...
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1answer
20 views

“projective maassbestimmung” in Automorphic Functions by Fricke + Klein

I was reading a copy of Fricke and Klein's Theory of Automorphic Forms, and I came across the phrase projective maassbestimmung in the first chapter. Google translate returns: maßbestimmung as ...
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1answer
77 views

Everyday life examples of hyperbolic rotations

I'm trying to find some intuition in Lorentz transformations. I understand that they are basically rotations by imaginary angle of vector of the form $\{ict,x\}$ (for $1+1$ space-time dimensions), and ...
0
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1answer
98 views

hyperbolic geometry proof with parallel lines

We are assuming hyperbolic geometry in this proof. Prove that for every line $l$ and external point P (im assuming point $P$ is not on line $l$), there are an infinite number of distinct lines ...
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0answers
192 views

Is there non-discrete group isomorphic to the fundamental group, what about the quotient?

It is known that (uniformization theorem) any Riemann surface can be written as the quotient of its universal cover by a discrete group (of Möbius transformations). This group is isomorphic to the ...
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0answers
75 views

Relations between Kleinian groups and quotient manifolds

In some specific situation there are some nice relations between a Kleinian group and its quotient manifold. For example, if $G$ is a once-punctured-torus group (i.e. a free subgroup of ...
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95 views

The lambda invariant

Consider the strip $\{x+iy: -1\leq x < 1 , y>1/2\}$ in the complex upper half plane and let $\lambda$ be the usual $\Gamma(2)$-invariant modular function on the complex upper-half plane. ...
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0answers
74 views

For an elliptic curve E, does there exist a cofinite Fuchsian group without elliptic elements with quotient E minus a finite subset

Let $E$ be a compact Riemann surface of genus 1, i.e., an elliptic curve. Let $P$ be the identity element of $E$. Question 1. Does there exist a cofinite Fuchsian group (or a Fuchsian group of the ...
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35 views

Derivatives of hyperbolic functions and Osborne's rule.

I am slightly confused when it comes to Osborne's rule when you take derivatives of hyperbolic functions. For example. The derivative of cotx is -cosec^2x, so there is a product of sines. So should ...
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0answers
21 views

Identification of polygon edges

In Klein's famous example of regular 14-gon made of 336 copies of (2,3,7) triangles, he used identification for edges such that side 2i+1 is identified with side 2i+6 (mod 14). But I wonder how could ...
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0answers
26 views

Find the equation of the conjugate of the hyperbola $xy+4-4x-2y=0$

Problem : Find the equation of the conjugate of the hyperbola $xy+4-4x-2y=0$ My approach : Solution : After simplifying the given equation of the hyperbola $(y-4)(x-2)=4$ $\Rightarrow $ ...
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0answers
25 views

curves in Poincare half space (3 dimensional hyperbolic geometry)

Okay maybe I am going a bit ahead of my self The Poincare half plane still has many mysteries for me But still I was puzzeling about the 3 dimensional variant of it. So lets assume an hyperbolic 3 ...
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0answers
18 views

Collinear Points and Congruent triangles

How might one show that three points are collinear? I am in hyperbolic geometry and am showing that two parallel lines also have a common perpendicular. I have two parallel lines $m$ and $l$ cut by a ...
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0answers
30 views

signature of a 2-dim linear subspace

Prop: If $U$ is a 2-dim linear subspace of $\mathbb{R}^{n,1}$ (Eculidean space with Lorentz-scalar product) with $U \cap H^n \neq \emptyset$, then the restriction $<\cdot ,\cdot>|_{U}$ has ...
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16 views

Decision Boundary of A Single Perception with Logistic Function

I am currently studying neural networks and have been trying to reason about this for a while to no avail. I understand that given a perceptron(such as above) with f as a step function, any ...
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0answers
38 views

Completion of hyperbolic 3-manifold (Thurston's lecture note)

I need help with a part of Thurston's lecture note. p.55~56 in the 4th lecture note states that when obtaining hyperbolic manifold by gluing polyhedra having some ideal vertices, the set of ...
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0answers
28 views

Pasch's axiom in the hyperbolic plane

Q: I am asked to show that pasch's axiom holds true in the hyperbolic plane. Pasch's axiom Idea: I was thinking of about a circle $\Gamma$ whose centre is the origin of the complex plane. ...
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23 views

On: geometry ; incidence axioms and a given set of points and straight lines

I need help on the following problem set: Let $P = \{ A, B, C, D, E \}$ be a set with five elements and let $$ \mathfrak{g} := \left \{ ...
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25 views

Prove by showing that CW is asymptotic parallel to DZ in the direction of W and Z

If XABY is a biangle and WCDZ is a figure formed so that AB=CD. angle BAX = angle DCW, and angle ABY = angle CDZ then WCDZ is a biangle. If Two lines AX and BY are asymptotic parallel in the ...
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52 views

HyperCube questions

I have three hypercube questions. 1) How many nodes does a d-dimensional HyperRing have (as a function of d) ? 2) How many edges ? 3)What is the degree of each node in a HyperRing with n nodes ? I ...
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14 views

constant $K$ and $k_g$ ovals growth

Referring to my recent post: Ovals of constant $ k_g$ on constant $K$ surfaces, using geodesic polar coordinates with radial geodesic lines built along v=constant around a fixed point on a constant ...
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55 views

Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
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47 views

Why does $\mathrm{Aut}(\mathbb{H}) = \mathrm{Isom}^{+}(\mathbb{H})$?

Suppose $T \in \mathrm{Isom}^{+}(\mathbb{H})$ . With out loss of generality we may assume that $T$ fixes two points $P,P′$ on the imaginary axis $i\mathbb{R}$. Now let $Q \in \mathbb{H}$. Since ...
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36 views

How to get the smallest sample size with max probability in Hypergeometric Distribution

A body of students has 30 male students and 20 female students. Suppose a sample of n students are drawn from this population. What is the smallest n that can yield the maximum probability to have 5 ...
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49 views

boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on ...
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31 views

Enlightening explanation of a theorem of Zimmert's

I'd like to know wether anyone has ever read an enlightening explanation (e.g. with geometric argument) of the following paper: Zimmert, R. Zur $SL_2$ der ganzen Zahlen eines imaginär-quadratischen ...
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109 views

Sum of angles in a hyperbolic triangle with one ideal angle

I want to calculate the sum of the angles of the triangle formed in the hyperbolic plane from the points $(-1,1), (0,1)$, and $(1,1)$. This forms an angle at the origin which has an infinite slope for ...