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

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Dirichlet Domain of a Fuchsian Group

Recall that a Fuchsian group is a discrete group $\Gamma \leq PSL(2, \mathbb{R})$ and that a Dirichlet domain for $\Gamma$ is a set $D \subset \mathbb{H}^2$ of the form $$\{z \in \mathbb{H}^2: d(z, p) ...
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2answers
67 views

Prove that a transformation from the hyperbolic group can not be loxodromic.

Prove that a transformation from the hyperbolic group can not be loxodromic. I know a loxodromic λ = kei$^\theta$ with k not equal to 1 and theta not equal to 0. But I'm unsure how to go after that, ...
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1answer
222 views

Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?

There is a sense in which all "interesting" properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations ...
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1answer
45 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 ...
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1answer
100 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 ...
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1answer
48 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 ...
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1answer
75 views

Simple understanding of convex co-compactness

I was looking for the definition of the term "convex co-compact" in simple cases. But most references I find are looking into a little bit sophisticated notions such as mapping class group, Schottky ...
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1answer
102 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 ...
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1answer
33 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
81 views

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides?

Can a hyperbolic quadrilateral have 3 obtuse angles and 3 equal sides? I have been trying to visually see if that is possible or not.
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1answer
33 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 ...
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1answer
49 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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1answer
42 views

Can I solve for the fractional volume of a hyperboloid?

This looks like a homework problem because it is. I'm stuck at the portion where I solve for fractional volumes. Suppose you are a part of a team designing a water tank in the shape of a hyperboloid. ...
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1answer
45 views

Quick Hypberbolic Geometry question concerning Saccheri Quadrilaterals

Can a Saccheri Quadrilateral have 3 congruent sides? I know the summit is less then the base, but could it happen that the base is the same length as the two vertical sides?
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34 views

conformal mapping

Prove that application ζ^-1:B^2---> H^2 is conform. ζ^-1(x1,x2,x3)=(x1/1+x3 , x2/1+x3) B^2 is unit disc and H^2 is hyperbolic 2-space and conformal mapping preserve angles.
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28 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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95 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 ...
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85 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. ...