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

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4
votes
6answers
225 views

Help me to remember $\operatorname{cosh}^{2}(y) -\operatorname{sinh}^{2}(y)=1$, some easy verification and deduction?

I can faintly visualize some way of deducing this formula with exponential functions but forgot it. How do you remember it? Suppose you just forget whether it is plus-or-minus there, how do you find ...
2
votes
2answers
370 views

Converting $\tanh^{-1}{x}$ to an expression involving the natural logarithm

I know how to convert $\sinh^{-1}{x}$ and $\cosh^{-1}{x}$ to $\ln{|x+\sqrt{x^2 \pm 1}|}$, but for some reason I am struggling to do the same for the following statement: $$\tanh^{-1}{\frac{x}{2}}$$ ...
4
votes
1answer
136 views

explicit isometry between metric spaces

Let $X=\mathbb{H}^2\times\left[0,1\right]$ (just $\left\{ (x,y,z)\big\vert y>0\right\}$ as a set) and consider the following two metrics: $$ds_1=\frac{dx+dy}{y}+dz$$ and ...
6
votes
2answers
2k views

Is an equilateral triangle the same as an equiangular triangle, in any geometry?

I have heard of both equilateral triangles and equiangular triangles. (For example, this sporcle quiz lists both.) Are these always equivalent, regardless of geometry? I know they are the same in ...
1
vote
1answer
248 views

Questions about Hyperbolic Isometries: The Standard Inversion

I have two questions regarding the inversion across the unit circle in the hyperbolic plane. Recall that the hyperbolic plane is a metric space consisting of the open half-plane $$\mathbb{H}^2 = ...
21
votes
9answers
3k views

What are the interesting applications of hyperbolic geometry?

I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous ...
2
votes
2answers
249 views

metric on the universal covering of a geometric manifold

We know that the universal covering of a closed hyperbolic 3-manifold can be identified with the hyperbolic space $\mathbb{H}^3$. Now, what is not very clear to me is how this identification has to be ...
0
votes
1answer
229 views

solving this laplacian

given the laplacian $ -y^{2}( \partial _{x}^{2} + \partial _{y}^{2} )f(x,y)=Ef(x,y) $ can we find a solution in the form $ f(x,y)= \sum_{n,m}C_{n,m} \phi (x,E) g(y,E)$ if we impose the extra ...
2
votes
0answers
49 views

limit set of Kleinian grouos 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 ...
1
vote
0answers
69 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 ...
4
votes
1answer
135 views

Backslash notation: $\Gamma {\setminus} \mathbb{H}^n$

I encountered this notation in a paper by Carron: When X = $\Gamma{\setminus}\mathbb{H}^n$ is a real hyperbolic manifold, ... $\Gamma$ is a discrete torsion free subgroup of SO$(n,1)$. My ...
5
votes
2answers
322 views

Symbolic coordinates for a hyperbolic grid?

Rephrasing     (one year later)    (original question is below) Apparently the original question wasn't clear, or nobody knows an answer (or both). So I will try to rephrase it. Look at your ...
18
votes
1answer
541 views

How to create mazes on the hyperbolic plane?

I'm interested in building maze-like structures on the [5, 4] tiling of the hyperbolic plane, where by maze-like I mean something akin to a spanning tree of the underlying lattice: a subgraph of the ...
3
votes
0answers
76 views

Construct tiling group from hyperbolic polygon

Given a hyperbolic $4n$-gon $P$ in the Poincaré disk, how can we construct explicitly the subgroup $G < \mathrm{Aut}{\left(\mathbb{D}\right)}$ which gives a tiling of $\mathbb D$ with fundamental ...
1
vote
1answer
148 views

Why is this fundamental group a discrete subgroup in $\operatorname{SL}_2(\mathbf{R})$ of finite volume?

Let $B$ be a finite set in $\mathbf{P}^1(\mathbf{C})$. Let $G$ be the fundamental group of $\mathbf{P}^1(\mathbf{C}) - B$. We can view $G$ as a subgroup of $\mathrm{SL}_2(\mathbf{R})$. Why is $G$ ...
1
vote
2answers
160 views

Perpendicular in conformal disk model

Firstly, please note that the related question can also be found at mathoverflow. The question is stated as following: In Euclidean Geometry, we know that from a given point there is an unique line ...
2
votes
0answers
188 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry?

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, then is there a similar thing do happen in Spherical and hyperbolic spaces? In ...
5
votes
1answer
631 views

Shortest path on hyperboloid

On the sphere $S^2$, the shortest path between two points is the great circle path. How about $H^2$, the hyperboloid $x^2+y^2-z^2=-1, z\ge 1$, with the Euclidean distance? Is there a formula for the ...
0
votes
1answer
82 views

Holomorphic function on an open subset of the complex upper-half plane

Let $f:\mathbf{H}\to \mathbf{C}$ be a holomorphic function on the complex upper-half plane and let $U$ be a bounded open subset in $\mathbf{H}$ contained in $$\{\tau \in \mathbf{H}: \mathrm{Im}(\tau) ...
0
votes
1answer
234 views

Nice formulas for the lambda invariant of an elliptic curve

Where can I find some nice formulas for the lambda invariant of an elliptic curve? I vaguely recall there's a nice product formula in terms of $q$, but a google search didn't give me much. Also, are ...
0
votes
0answers
90 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. ...
3
votes
3answers
153 views

A question about hyperbolic functions

Suppose $(x,y,z),(a,b,c)$ satisfy $$x^2+y^2-z^2=-1, z\ge 1,$$ $$ax+by-cz=0,$$ $$a^2+b^2-c^2=1.$$ Does it follow that $$z\cosh(t)+c\sinh(t)\ge 1$$ for all real number $t$?
2
votes
0answers
41 views

ruling out non Pseudo-anosov automorphisms

We are given a fibration $S\to M\to S^1$ where S is a compact hyperbolic surface, M a 3-manifold and $S^1$ the circle. Topologically speaking, it is clear that M has to be the mapping torus ...
1
vote
1answer
97 views

A question on hyperbolic geometry

I am reading a book that seems to claim the following. I suspect that there may be a misprint, or some assumptions missing. For $A=(x_1,y_1,z_1),B=(x_2,y_2,z_2)\in R^3$, define $$\langle ...
2
votes
1answer
115 views

Real elliptic curves in the fundamental domain of $\Gamma(2)$

An elliptic curve (over $\mathbf{C}$) is real if its j-invariant is real. The set of real elliptic curves in the standard fundamental domain of $\mathrm{SL}_2(\mathbf{Z})$ can be explicitly ...
2
votes
0answers
167 views

The fundamental group of the mapping torus is doubly degenerate

Consider an hyperbolic compact surface $S$ (hence with genus $>1$) and a Pseudo-Anosov diffeomorphism $\varphi\colon S\to S$. We call "mapping torus" the 3-manifold ...
3
votes
1answer
106 views

A Kleinian group has the same limit set as its normal subgroups'

It should be well known that a Kleinian group and all its normal (non-elementary) subgroups have the same limit set. Do you know any book/article where I could find the proof? Thank you.
0
votes
1answer
200 views

Fundamental group of a convex co-compact surface

Let $G \subset SL_2(\mathbb R)$ be a free subgroup generated by a symmetric set of generators $\{ a_1^{\pm 1},\ldots,a_n^{\pm 1} \}$ such that the action of $G$ on the upper-half plane $\mathbb H$ in ...
4
votes
1answer
392 views

The law of sines in hyperbolic geometry

What is the geometrical meaning of the constant $k$ in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}=k$ in hyperbolic geometry? I know the meaning of the ...
3
votes
2answers
388 views

Coordinates and distance in higher dimensional spherical and hyperbolic space

For n-dimensional spherical space, it seems to me the representation of points is easiest and most manipulable as unit vectors, with distance being the vector dot product (which is the cosine of the ...
2
votes
3answers
272 views

Gromov boundary — TFAE

I am a newcomer to hyperbolic geometry and was trying to understand some of it in the context of dynamics, for reading certain literature. Let a discrete subgroup $G$ of $SL_2(\mathbb R)$ act on the ...
8
votes
1answer
279 views

Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?

Motivation I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points ...
4
votes
1answer
298 views

Models of hyperbolic geometry

Wikipedia states the following: [The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
1
vote
0answers
71 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 ...
4
votes
1answer
230 views

Difference between a hyperbolic line and a geodesic

The setting for hyperbolic space in this question will be the upper half plane. Now I know that to measure the distance between two points $p$ and $q$ in the upper half plane, we take $ \inf ...
1
vote
1answer
212 views

Relation between Hilbert theorem and pseudosphere (also called hyperbolic plane or Bolyai–Lobachevsky plane)

The Hilbert theorem states that there exists no complete regular surface S of constant negative gaussian curvature $K$ immersed in $R^3$. Ok.. so I'm guessing that the surface of revolution of the ...
2
votes
1answer
118 views

Point on a surface with no geodesics passing through

Take an orientable surface $S_g^s$ of genus $g$ with no boundaries and $s$ points removed and fix a complete hyperbolic metric of finite area (assuming that the Euler characteristic allows an ...
3
votes
2answers
646 views

Geodesic Uniqueness in the Hyperbolic Plane

I am studying Hyperbolic Geometry. At this part, I have proved that semicircles and straight lines orthogonals to the real axis are geodesics in the hyperbolic plane. But how I proof that this ...
2
votes
1answer
77 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 ...
2
votes
1answer
314 views

Proving an equality for an equilateral triangle in the Poincare model

I've been working a good while trying to establish an equality, but have made little success. Suppose you're working in the Poincare disk model inside an ambient Euclidean plane. If an equilateral ...
4
votes
4answers
258 views

Completeness of Upper Half Plane

I am trying to prove that the upper half plane, defined as $\mathbb{H} = \{z \in \mathbb{C} : \Im(z)>0 \}$, is complete with respect to the hyperbolic metric. First I note that if I have some ...
10
votes
2answers
189 views

Embedding the Infinite Binary Tree in Regular Tilings

Consider the regular tiling $(m,n)$ in which $m$ $n$-agons meet at each vertex. Most of the time this tilings have to "live" in the hyperbolic plane. The edges of its polygons define a graph where two ...
3
votes
1answer
159 views

Möbius Transforms that preserve $\mathbb{H}$

I know that every möbius transform that preserves the upper half plane is of the form $m(z) = \frac{az+b}{cz+d}$, where $a,b,c,d \in \mathbb{R}$, or $m(z) = \frac{a\bar{z} + b}{c\bar{z} + d}$, where ...
3
votes
1answer
1k views

Möbius Transforms that preserve the unit disk

Say I wish to prove that every möbius transformation of the unit disk onto itself can be written in the form $A(z) = e^{i\theta}\frac{z+a}{1+\bar{a}z}$, where $\theta$ is a real number and $a$ is a ...
1
vote
1answer
634 views

Deriving the distance between two distinct points on the Upper Half Plane $\mathbb{H}$

I am trying to derive the distance between two arbitrary points in hyperbolic space; the model I'm using is the upper half plane model. So the distance is just $\int_f \rho(z) dz$, where $\rho(z) = ...
5
votes
1answer
486 views

Interpretation of Hyperbolic Metric and Möbius Transforms

I was wondering if someone could explain the interpretation of the following results. In hyperbolic geometry, we say that lengths are invariant under the action of Mob($\mathbb{H}$) if given any ...
0
votes
3answers
367 views

Exciting Topics in Hyperbolic Geometry

I am a first year student and a learner of hyperbolic geometry. I was wondering if you could suggest some exciting topics to research about in this field (some people suggested fundamental polygons ...
8
votes
3answers
484 views

Simulation of Brownian Motion

If I want to simulate Brownian motion in the Euclidean space I can simulate it by a point that is moving a distance $\epsilon$ in an arbitrary direction then it randomly choose a new direction and ...
1
vote
1answer
262 views

Hyperbolic area and $SL_2$

Given that $\mu(A) := \iint_{A}\frac{\mathrm dx\mathrm dy}{y^2}$ where $A \subset H$ and $H$ is the upper half-plane, I need to show that: a. The measure $\mu$ is invariant under all $g \in ...
4
votes
1answer
277 views

Circle preserving homeomorphisms in the closure of $\mathbb{C}$ and Möbius Transformations

I am presently a learner of Hyperbolic Geometry and am using J. W. Anderson's book $Hyperbolic$ $Geometry$. Now the author presents a sketch proof of why every circle preserving homeomorphism in ...