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

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1answer
68 views

Limit sets of representations of once-punctured torus groups and circle packings

Let $\rho\colon\pi_1(T_1)\to PSL(2,\mathbb{C})$ be a faithful representation of the fundamental group of a once-punctured torus. If both the components of the convex core in the quotient manifold are ...
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1answer
307 views

Hyperbolic coordinates

You can uniquely specify any point in 2D Euclidian space using 2 numbers: the distance from the infinitely long X-axis, and the distance from the infinitely long Y-axis. How do you uniquely specify a ...
4
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1answer
115 views

Fuchsian groups and surfaces

It's a fact that if two Fuchsian group are conjugate, the corresponding surfaces are isometric. Is the converse true ? Take 2 isometric Riemann surfaces $S$ and $S'$(which are covered by the upper ...
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0answers
84 views

Distance realizing geodesic in a hyperbolic surface

Suppose $S$ is a hyperbolic surface with geodesic boundary and $P$ is a hyperbolic pair of pants with $a$, $b$, $c$ geodesic boundary. Let $\gamma$ be the unique geodesic realizing the distance ...
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2answers
449 views

How does a conformal mapping preserve angles in hyperbolic geometry?

Suppose I have a sector $D = \{0 < \arg z < \alpha\}$ where $\alpha \leq 2\pi$. If I apply the function $w = \frac{\zeta - i}{\zeta + i}$ from the upper half plane to the unit disc ($\zeta = ...
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1answer
183 views

Geodesic distance on complex upper half plane

Let $z$ be a point in a fundamental domain of $\Gamma(2)\subset \mathrm{SL}_2(\mathbf{Z})$ in the complex upper half plane. Does there exist an $\epsilon >0$ such that the geodesic distance ...
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3answers
1k views

how to generate tesselation cells using the Poincare disk model?

I'm a computer programmer, and while I like math, this is an area where my understanding of math falls short of what I need in order to apply it successfully. I've been looking at M.C. Escher's ...
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1answer
481 views

Generalized Laws of Cosines and Sines

I wonder the "laws of sines and cosines" in the two cases below and how to derive them. (or any related sources) (i) For geodesic triangles on a sphere of radius $R>0$. (so constant curvature ...
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1answer
159 views

A book to study about hyperbolic plane, hyperbolic translations, etc.

In this paper, page $6$, the authors state the following: The translations of the hyperbolic plane are defined as products of two central symmetries; the set of hyperbolic translations forms a ...
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1answer
196 views

When does there exist an isometry that switches two subspaces?

Let $V$ be a real vector space of finite dimension and let $\langle \cdot, \cdot \rangle$ be a non-degenerate symmetric bilinear form on $V$. Let $U, W \subseteq V$ be linear subspaces such that ...
3
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1answer
259 views

Thrice-punctured sphere

This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45: the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric. I ...
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1answer
184 views

Hyperbolic triangle and two points in Poincare disk

Given a hyperbolic triangle $T$ and two points $p$ and $q$ in Poincare disk. Note that $p$ and $q$ are outside the triangle. If $p$ has shorter distances to the three vertices of $T$ than $q,$ can we ...
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1answer
125 views

Bounding the number of integer solutions of the following inequality

Let $r\geq 1$ be a real number, $-1\leq x\leq 1$ a real number and $y>2$ a real number. We consider this data to be fixed. How can I obtain an upper bound on the number of $(a,b,c,d)\in ...
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1answer
114 views

Geometry of a hyperbolic triangle

How can we find the upper bound of the hyperbolic distance from any point on the side $AB$ to either $AC$ and $BC$ for the hyperbolic triangle $\triangle ABC$? Help will be appreciated.
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0answers
169 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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1answer
181 views

Wikipedia article on Hyperbolic geometry

I was reading the Wikipedia article on hyperbolic geometry and have come across the line geodesic paths are described by intersections with planes through the origin Why is this necessarily ...
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1answer
175 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
142 views

Why are perpendicular bisectors 'lines'?

Given two points $p$ and $q$ their bisector is defined to be $l(p,q)=\{z:d(p,z)=d(q,z)\}$. Due to the construction in Euclidean geometry, we know that $l(p,q)$ is a line, that is, for $x,y,z\in ...
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1answer
185 views

Real part of $\sinh(z)$

Is it true that for every $z \in \mathbb{C}$ $$\begin{align} &2\mathfrak{Re}(\sinh z) &= \sinh z + \sinh \bar z\\ &2i\cdot \mathfrak{Im}(\sinh z) &= \sinh z - \sinh \bar z ...
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2answers
469 views

Does anyone know a good hyperbolic geometry software program?

We are currently using this program called NonEuclid but it is a little frustrating to use sometimes and I was wondering if anyone knows another program for hyperbolic geometry.
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0answers
202 views

the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
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6answers
217 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 ...
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2answers
336 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: $$\tanh^{-1}{\frac{x}{2}}$$ ...
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1answer
132 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 ...
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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 ...
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1answer
238 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 = ...
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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 ...
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2answers
213 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 ...
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1answer
227 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 ...
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0answers
42 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 ...
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0answers
62 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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1answer
120 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 ...
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2answers
301 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
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1answer
516 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
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0answers
72 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 ...
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1answer
141 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 $G$ ...
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2answers
158 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 ...
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0answers
182 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 ...
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1answer
561 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 ...
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1answer
81 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) ...
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1answer
207 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 ...
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0answers
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. ...
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3answers
150 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 $t$?
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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 ...
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1answer
92 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 ...
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1answer
114 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 ...
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0answers
165 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
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1answer
103 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.
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1answer
192 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 ...
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1answer
364 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 ...