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

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4
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1answer
148 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 ...
7
votes
2answers
413 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 ...
20
votes
1answer
635 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
83 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
160 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
179 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 ...
3
votes
1answer
238 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 ...
7
votes
4answers
909 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
92 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
292 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 ...
1
vote
0answers
97 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
154 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
58 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
100 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
137 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 ...
3
votes
0answers
191 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
120 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
226 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 ...
5
votes
2answers
481 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 ...
4
votes
2answers
467 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
327 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 ...
9
votes
1answer
328 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
2answers
324 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
76 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
263 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
245 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
129 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
860 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
80 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
360 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
279 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 ...
12
votes
2answers
200 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
183 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 ...
4
votes
1answer
2k 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
719 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
559 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 ...
1
vote
3answers
411 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
538 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
304 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
299 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 ...
10
votes
1answer
332 views

Teichmüller spaces via representations

I don't have much expertise in this area but I am confused by a remark I overheard regarding Teichmüller spaces. I was always under the impression that for a surface $S$ (say genus $\geq 2$) ...
3
votes
1answer
538 views

Hyperbolic metric on the torus?

Here is a silly mistake I am making: where exactly is the mistake? I know that torus cannot hold a metric of constant curvature -1 ( hyperbolic metric ). But what if I do this: The upper ...
0
votes
1answer
99 views

Question on proof in “Primer on MCGs”

This is a question about the proof of Proposition 1.4 in Farb and Margalit's "Primer on Mapping Class Groups" (in v. 5.0, it is on page 37 in the PDF, which you can download here). The proposition ...
1
vote
1answer
118 views

What is the cardinality of a subset of the hyperbolic upper half plane?

Given a subset of the hyperbolic upper half plane, say an ideal triangle (so with vertices on the boundary), what is the cardinality of all points contained in the interior?
4
votes
2answers
840 views

Finding Möbius transformation from fixed points

Given a non-parabolic transformation which is also an orientation preserving isometry in the hyperbolic upper half plane union the boundary, if I know the two fixed points and they are two different ...
5
votes
2answers
126 views

Two hyperbolic surfaces corresponding to conjugate Fuchsian groups are isometric

I have a basic question : a) Suppose $\gamma $ and $ \gamma' $ be conjugate Fuchsian groups acting freely and properly discontinuously on the upper half-plane H to produce two Riemann surfaces $ ...
4
votes
2answers
277 views

How to analyze triangles in Lobachevsky geometry?

I got an assignment to prove certain things about right triangles in Lobachevsky geometry, but so far I don't know where to start. What model is the best for studying these objects? What is the ...
12
votes
3answers
526 views

What is the connection of the sequence 3, 4, 5/3, 2/3, 1 with deep topics?

Quote from Don Zagier (Mathematicians: An Outer View of the Inner World): " I like explicit, hands-on formulas. To me they have a beauty of their own. They can be deep or not. As an example, ...
1
vote
1answer
288 views

Geodesic on half-plane determined by tangent vector

The upper-half plane $\mathbb H$ carries a hyperbolic metric and the geodesics are semicircles with base on the real line. We consider oriented geodesics. Let $x \in \mathbb H$ and let $v$ be a unit ...
5
votes
1answer
626 views

Hyperbolic geometry. 3 dimensions. What is not well understood?

According to Mathworld, hyperbolic geometry is well understood in 2 dimensions but not in 3 dimensions. http://mathworld.wolfram.com/HyperbolicGeometry.html What isn't well understood about ...