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

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The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g-6$ geodesic length functions

Setting: It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ boundary components (satisfying $2g + b \geq 3$) can be ...
6
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How to parameterize these pretty hyperbolic (Amsler) surfaces?

I've seen the attached images describing surfaces of negative curvature. I was wondering if there exist such surfaces with constant Gaussian negative curvature. To this end, I attempted to model the ...
5
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80 views

Compact surfaces with boundary of constant negative curvature

Consider a surface (with boundary) diffeomorphic to $S^1 \times [0, 1]$ and with constant negative curvature, sitting inside $\mathbb{R}^3$. All the examples I know of such surfaces are "part of" (or "...
5
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199 views

Fundamental solution to Laplace equation on arbitrary Riemann surfaces

So, I've seen in a few places this method of calculating the heat kernel on a manifold given the kernel of its universal cover, through a so-called 'tiling method' as in section five of this paper (...
5
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118 views

Tilings of the Hyperbolic plane

Given a tiling of the hyperbolic plane projected onto a unit disc such as this which can be considered as a graph. I then define some functions: $f(r) =$ number of graph nodes contained within the a ...
4
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41 views

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of $...
4
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46 views

Misinterpretations of Hilbert's Theorem?

I've seen a few posts here that make certain claims that are related to Hilbert's theorem. For instance: "I know that there is no complete surface embedded in $\Bbb R^3$ of constant curvature $-k$ ...
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710 views

Curvature of Hyperbolic Space

I'm trying to prove that hyperbolic space has constant sectional curvature $-1$, but keep running into difficulties. Could someone show me a way out? I've been given the metric $$g_{ij}=\frac{4\...
4
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201 views

How do we define a complete metric on a Riemann surface with punctures?

This question is related to another question. If we have a Riemann surface with punctures of negative Euler characterstisc, how can one define a complete hyperbolic metric? I know that in this ...
4
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520 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 y>0\...
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Estimating Lorentzian inner product

Let $\mathbb{L}^{n+1}$ be the Lorentz space, that is, the Euclidean space $\mathbb{R}^{n+1}$ equipped with the nondegenerate bilinear form $$ \langle x, y\rangle = x_1 y_1 + \cdots + x_n y_n - x_{n+1}...
3
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110 views

Zeta functions for flows on hyperbolic surfaces

The Riemann zeta function is defined by $$ \zeta(s)=\sum_{n=1}^\infty \frac1{n^s} $$ and can be written in the form $$ \zeta(s)=\prod_p\frac1{1-p^{-s}}, $$ where the product is over all prime numbers $...
3
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49 views

What are some good graphics programs for depicting hyperbolic geodesics?

I'm looking for some software that allows one to draw accurate pictures in hyperbolic space. In particular, I want to be able to specify pairs of points and generate the geodesic between them, in ...
3
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52 views

Representation of hyperbolas.

I am well aware of the matrix representation for rotation of points on a circle with reals $$M_{R(\theta)} = \left(\begin{array}{rr} \cos(\theta) & \sin(\theta) \\ -\sin(\theta) & \cos(\theta)\...
3
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66 views

Parallel postulate in hyperbolic geometry question

Let $L =\{y=0\}\cap\mathcal{H}^{2}$, and let $P=(3,2,2)$. Show that the parallel postulate fails in $\mathcal{H}^{2}$ by giving two lines $L',L'' \in \mathcal{H}^{2}$ with $P\in L',L''$ and $L\cap L'=...
3
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52 views

Finding a fundamental polygon for two-generator subgroup of PSL(2,R)

Suppose we are given two hyperbolic isometries $A$ and $B$ with intersecting axes. Assume also that the commutator $\left[A,B\right]$ is an elliptic element (perhaps of infinite order). I would like ...
3
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100 views

Question regarding the projective models of the anti-de-Sitter spaces and good online references for learning them from the scratch? (Specifics below)

As my title says above, I am trying to find answers to and also good online reference where I can find complete description of projective models of hyperbolic space, de-Sitter space and anti-de-Sitter ...
3
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183 views

Axis of the product of two loxodromic isometries

Suppose that $X$ and $Y$ are two loxodromic isometries of the hyperbolic space and that the product $XY$ is also a loxodromic element. We consider the axes of these three elements. I'd like to know if ...
3
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58 views

reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result: For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
3
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89 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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202 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 $$M=M(\varphi)=S\times\left[0,1\...
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15 views

Poincaré cylinder

The Poincaré disk model of the hyperbolic plane is the open disk ${\rm int}(D^2)$ with a certain metric $d_H(x,y)$. What happens if I take the open tube ${\rm int}(D^2)\times\Bbb R$ with the metric: $...
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27 views

Seeking proof to a Hyperbolic polygon conjecture

In the course of writing a(n Honours) thesis, I'm searching for a proof to a conjecture that appears very likely to be true. Many results will rely upon it. My own attempts to prove it have been ...
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28 views

Dirchlet region for the Hecke Triangle group

Let $G_n$ for $n>2$ be the subgroup of $SL_2(\Bbb R)$ generated by $$ \begin{bmatrix} 0 & -1\\ 1 & 0 \\ \end{bmatrix} \ \text{and} \ \begin{...
2
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31 views

Distance between points lying on a hyperbola?

The question is rather simple but I can't find the answer I'm looking for anywhere. On an ordinary 1-dimensional hyperbola, given two points on the hyperbola, what is the length of the path between ...
2
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44 views

Harmonic maps between Riemann surfaces

In 'Compact Riemann surfaces' Jost defines harmonic maps between surfaces $S_1,S_2$, with local coordinates z on $S_1$ and metric $\rho^2|du\,d\overline{u}|$ on $S_2$ as $u\in C^2$ solving the ...
2
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27 views

congruency of triangles in hyperbolic and spherical geometry

In Euclidean geometry, we have the following congruencies of triangles: side-side-side, side-angle-side, angle-angle-side = angle-side-angle (because of the angle sum) and side-side-angle (only if the ...
2
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32 views

Subgroups of $\text{PSL}(2, \mathbb{R})$ Closed under Transposition

I am wondering, does anyone know if there is a classification of transposition-closed (Fuchsian) subgroups of $\text{PSL}(2, \mathbb{R})$? I can't read French, so for all I know it's sitting in the ...
2
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38 views

another pythagorean theorem in hyperbolic geometry

on https://en.wikipedia.org/wiki/Pythagorean_theorem#Non-Euclidean_geometry it says However, the Pythagorean theorem remains true in hyperbolic geometry and elliptic geometry if the condition ...
2
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77 views

Villarceau circle as a Loxodrome

A circular Clifford torus (radius at flat circle = h, section radius $ a , a<h $ ) is cut by a plane at an angle $ \cos \alpha = a/h \tag{1} $ centrally to the symmetry axis, the line of ...
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40 views

Example of a doubly degenerate Kleinian group which does not come from a mapping torus

Doubly degenerate Kleinian groups are discrete subgroups of $PSL(2,\mathbb{C})$ whose limit set is all of $S^2$, the boundary of $\mathbb{H}^3$. A standard example of such a group is given as follows:...
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111 views

Linear Isoperimetric Inequality is invariant under quasi isometry

Suppose $X$ and $Y$ are quasi-isometric. Show that $X$ satisfies a linear isoperimetric inequality iff $Y$ satisfies a linear isoperimetric inequality. My idea: Suppose $X$ satisfies a linear ...
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35 views

Corollary of Wolpert lemma

Recall Wolpert Lemma. Let $S$ be a surface with genus greater than 2, let $[X,f]$ and $[Y,g]$ two points of $T(S)$ (Teichmüller space) and let $\phi \colon X \to Y$ a $K$ quasi conformal homeo. Then ...
2
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43 views

Homeo- and diffeomorphism groups of oriented surfaces

I'm interested in the structure of homeo- and diffeomorphism groups of oriented surfaces, especially in hyperbolic case. For example, does the homeomorphism group retracts on the diffeomorphism group (...
2
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79 views

constructing an equilateral triangle in the Beltrami klein model

I am puzzeling with the following: Using the beltrami klein disk of hyperbolic geometry (see https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein_model ) (PS not the poincare disk model) and given ...
2
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79 views

Calculating hyperbolic length

So, I am looking at a question and I'm having a hard time solving it. So I know the formula but my question is first, what is $\alpha$ and what is $\beta$? So I calculated some values: I found $...
2
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43 views

Compute hyperbolic length of the arc of the circle

Compute the hyperbolic length of the arc of the circle $ x^2 + y^2 = 25$ that lies between (3, 4) and (4, 3). From my notes I know the formula is $$ \ln \frac{{\csc \beta - \cot \beta }}{{\csc \...
2
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63 views

Thread constructions in the Poincaré's disc

I just came across the following image (source) and realised something that should have been obvious to me a while ago: it should be possible to construct the envelopes of curves in the Poincaré disc ...
2
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59 views

Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that $...
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0answers
250 views

What is a manifold with cusp?

What I am primarily currently learning about is hyperbolic geometry and methods to find hyperbolic structures on triangulated manifolds. I see phrases such as 'cusp ends' and 'manifold with one cusp' ...
2
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53 views

Submanifold of a Kobayashi hyperbolic manifold

Let $M$ be complex manifold which is Kobayashi hyperbolic. Let $N$ be a submanifold of $M$ obtained as the zeroes of an analytic submersion $f : M \rightarrow R$, $R$ complex manifold. Question : ...
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66 views

Big picture question about the Thurston Metric on Teichmüller Space

I'm having difficulty in appreciating the significance of the Thurston metric on Teichmüller space. It seems like a lot of work to develop the theory of this asymmetric metric space with fewer ...
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136 views

What is the Area formed when a line is traced between two 3D curves?

This question is quite related to intersection of cylinders, Hyperbolic paraboloid and modelling. I am welding a trunnion to a pipe (both are hollow cylinders in different geometry). They intersect ...
2
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24 views

software to decide whether a 2-generator subgroup of PSL(2,R) is discrete/free

Gilman developed an algorithm with polynomial complexity that, given two elements in PSL(2,R), decides whether the group they generate is free/discrete or not. I was wondering whether anybody ever ...
2
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98 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 ...
2
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64 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 $M=M(\...
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16 views

Geometry - Inversion/Cross Ratios

Problem 5. Let ABCD be a given convex quadrilateral with sides BC and AD equal in length and not parallel. Let E and F be interior points of the sides BC and AD respectively such that BE = DF. The ...
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24 views

Showing that the inversion of a circle can be written in a certain way

Let $\phi = \dfrac{1}{r} ∘ T_{-O_C}$, $O_C$ the center of $C$, $T_{-O_C}$ is a translation. I want to show that the inversion of a circle $C \in \mathbb{C}$ can be written as: $$\iota_C = \phi ∘ \...
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Hyperbolic isometries and finite order elements

I'm reading a paper and I'm uncertain about one of its claims. I was wondering if someone could clarify. Namely, it states that for a discrete subgroup of $\text{Isom } H^n$, the finite order elements ...
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29 views

Can the hyperbolic orbifold 2*55 be smoothly and isometrically embedded in 3-space?

Grow a square in the hyperbolic plane until its vertex angles become $\pi/5$. Assuming that the constant Gaussian curvature of our hyperbolic plane is $-1$, the sides of the resulting hyperbolic ...