Questions tagged [hyperbolic-geometry]
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885 questions
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Quadrilateral fundamental domain
Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find the angle sum ...
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Geodesic curvature in hyperbolic geometry- Poincaré disk model [closed]
How is geodesic curvature defined with an ODE or geometric construction in the hyperbolic plane?
How do hyperbolic geodesics change when there is deviation from geodesy? How to construct or graph ...
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How to draw these triangles in hyperbolic geometry? [closed]
EDIT1:
In what follows I am pre-pending some omitted considerations regarding intersection of two small circles on a sphere resulting in two diangles, referring to them as the minor and major ...
- 917
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How was the pair of pants introduced [closed]
There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful?
For example, does it have any application if we estimate the perimeter or ...
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delta hyperbolicity of a product of poincare balls? [closed]
I know how to compute the $\delta$-hyperbolicity of a $2D$ poincaré disk of radius 1, but I was wondering how to generalize such computations to:
Higher dimensions
Poincaré disk with radius $r$ and ...
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Lifts of geodesics on surfaces onto the universal cover [closed]
self-intersecting geodesic on hyperbolic surface of genus 2
Given a self intersecting geodesic on a hyperbolic surface of genus 2 as in the picture, how can I understand precisely what the lift to ...
- 11
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Real section of moduli space of Riemann surfaces
In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset ...
- 1,435
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Annuli and pinched annuli vs circles and horocycles
Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed.
If we want to parametrise annuli with punctures at one boundary, give the punctures ...
- 1,435
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Cutting a circle from the hyperbolic plane
Let D be the Poincare' disk its natural hyperbolic metric and with at least 1 marked point on $\partial D$. Suppose I cut an hyperbolic circle of radius $r$ away from it, then I get a Riemann surface ...
- 1,435
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Topological entropy and pseudo-Anosov dilatation for punctured surface
Let $S_{g,n}$ be a closed surface of genus $g$ with $n$ punctures. Assume $2-2g-n<0$. Let $f$ be a pseudo-Anosov mapping class with dilatation $\lambda_f$. In the introduction (1st page) of the ...
- 1,713
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Generalized McShane Identity for Closed Riemann Surfaces
There is an identity for the hyperbolic Riemann surfaces with at least one border. The identity is known as Generalized McShane Identity or Mirzakhani-McShane Identity proved by Mirzakhani in her ...
- 989
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Fuchsian groups and surface groups
The following question may be trivial or inappropriate; I am not sure though.
It is known that a cocompact oriented Fuchsian group $\Gamma$ admits a presentation: for given $m,g,d_i \geq 0$
$$
\...
- 601
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Schwarz–Ahlfors–Pick theorem for hyperbolic pair of pants
Let $\mathcal{P}$ be a hyperbolic pair of pants with geodesic boundary and $\Sigma_3$ be the hyperbolic thrice punctures sphere. I want to construct a conformal map $f:\mathcal{P}\rightarrow \Sigma_3$ ...
- 1,713
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What interesting information can be deduced from knowledge of how deep a geodesic ventures into the cusp
First of all I have to apologise as I am not a geometer and my knowledge of geometry is poor. Let $M$ be the modular surface and $\gamma$ to denote a geodesic in $M$. In the the following paper by ...
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Change of length of curve when Fenchel-Nielsen length coordinate increase
Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...
- 1,713
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Comparison theorem for Lambert quadrilateral
A Lambert quadrilateral is a quadrilateral three of whose angles are right angles. And in 2-d hyperbolic space $\mathbb H^2$, we have nice formulas for the fourth angle.
If $AOBF$ is a Lambert ...
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Algorithm to generate hyperbolic metric on a compact surface
Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
- 1,713
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Immersed surfaces in Hyperbolic 3-manifolds
Given a hyperbolic 3-Manifold $M=\Gamma_{0}\setminus\mathbb{H}^3$, and a smooth, connected, compact immersed negatively curved surface $\Sigma=\Gamma\setminus\widetilde\Sigma\subset M$, where $\...
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Is triple point intersection 'generic' in Teichmuller space?
Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...
- 1,713
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Finding Riemannian metric explicitly from the conformal structure on a surface
Each Riemann surface structure $S'$ on a topological surface $S$ can be obtained from a Riemannian metric which looks like $ds^{2} = g_{11}dx^{2} + 2g_{12}dxdy + g_{22}dy^{2}$ in local coordinates ...
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Coarse geometry of minimal surfaces in non-positively curved manifolds
Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface.
(You can basically image $X$ as a ball and $S$ as an embedded disk whose ...
- 1,804
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How to pick out harmonics based on boundary conditions?
(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...
- 1,893
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Bounds on norm of harmonic function on degenerating hyperbolic surface
Suppose I have a Riemann surface with a shrinking geodesic which is degenerating towards a surface with a cusp, and I consider a neighborhood of the shrinking geodesic, $C_[a,b] = [a,b]\times \mathbb{...
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Measuring the complexity of a knot by minimum number of simplices to tile the complement
This is essentially a duplicate of: Lower bound on number of tetrahedra needed to triangulate a knot complement
Suppose a knot $K\subseteq\mathbb S^3$ is such that the complement $\mathbb S^3\...
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A question on Cayley graphs and hyperbolic 3-manifolds
There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
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Hyperbolic version of Sylvester co-linear problem
Is the hyperbolic version of Sylvester co linear problem true?
- 356
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Number of regions created by r hyper-planes in n-dimensional space [closed]
I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r)
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Why is a plane graph Delaunay realizable if stellating a face makes the graph inscribable?
I have come across the following lemma in several papers (for instance see lemma 2.2 in this) (and some authors state this the proof immediately follows from basic properties of the stereographic ...
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Hadamard submanifolds of $k$-fold product of hyperbolic plane
Let $\kappa>0$ and $d,k$ be a positive integers with $k\ge d$. For $k\in \mathbb{Z}^+$ large enough, can one find a geodesically complete and simply connected $d$-dimensional Riemannian ...
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Reference for 'Normal Subgroups of Fuchsian Groups'
I am looking for a reference on how to explicitly construct normal subgroups of a given Fuchsian group. I appreciate any help.
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Hyperbolic structures on infinite type surfaces
Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do ...
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What does it mean exactly for a pair of $S^0$'s to be unlinked on a knot $K$?
I am trying to learn about the effects of knot mutation on the hyperbolic manifolds obtained via hyperbolic Dehn surgery, and I'm currently reading Ruberman's paper "Mutations and Volumes in $S^3$" (...
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Geodesics under deck transformations on hyperbolic surface
I have a question regrading the deck transformations. As we know, for the 2-torus $\mathbb T^2$, if we have a geodesic $\widetilde\gamma$ on the corresponding covering space $\mathbb R^2$, and $\alpha$...
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Relations between automorphisms of field of rational functions and Mobius Transfomation
Proposition: If $F$ is a field, let $F[x]$ be the ring of all polynomials whose coefficients are in $F$. The fraction field of $F[x]$, denoted $F(x)$, is defined to be the ratios $r(x) = f(x)/g(x)$ ...
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Teichmuller Theory question : Beltrami forms on hyperbolic Riemann surfaces whose lifts are smooth upto the boundary of $\mathbb{D}$
Hello, my question is related to Teichmuller Theory. Let $D$ be the open unit disk and $X=D/{\Gamma}$ be a hyperbolic Riemann surface of the Fuchsian group $\Gamma$. In Teichmuller theory, we have ...
- 1,471
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Generator of translation for the hyperbolic plane? [closed]
What is the generator of translation in the Beltrami-Klein model of the hyperbolic plane?
- 1,532
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A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem
Hello,
I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...
- 1,471
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Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds
In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...
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What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?
I have been reading up on $\delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $\delta$-hyperbolic spaces and how the Gromov product is extended to it. Could ...
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If $i(x,z)\neq 0$ and if $y$ is conjugate of $x$, then what can we say about $i(x*y,z)$?
Let $S_g$ denote the closed oriented surface of genus $g\geq 2$. Let $x,y$ be two different (upto fixed base point homotopy) but freely homotopic curves, i.e. $y$ is a non-trivial element from a ...
- 3,751
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Entropy of Negatively pinched manifolds
Suppose $M$ is a compact negatively pinched Riemannian manifold of dimension $n$. We normalize the metric such that $-1\le K\le -a^2$ for some $0<a\le 1$. Let $G$ be the fundamental group of $M$. ...
- 1,101
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Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? [closed]
In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e.,
$$
S_{\Delta}=\frac{1}{2}a.h,
$$
where $a$ is the length of base and the $h$ is ...
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ideal triangles in a punctured torus
I googled it and wikipidead it too : but apparently there is no definition of an ideal triangle on a punctured torus ( i.e a compact [ hyperbolic ] surface with one genus and one boundary component, ...
- 1,471
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Geodesic whose one end is at a ideal point
We know that every hyperbolic manifold $M$ is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can we consider the inverse image, under the isometry, of the geodesic $\gamma$ that connects a ...
user519801
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Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul
In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
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When a polygonal line become a loop in hyperbolic plane?
Suppose we have a 5 tuple of positive real numbers $(l_1,l_2,m_1,m_2,m_3)$, with $m_i \in (0,\pi)$ for all $i$. Now fix a point $v_1$ in the hyperbolic plane. Then consider a geodesic of length $l_1$ ...
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Half space vs growing balls in the hyperbolic plane [closed]
Let $p$ be a point in the $2$-dimensional hyperbolic space $H_2$. Consider a normal coordinate system $(x,y)$ at centered at $p$. Let $o_x\in H_2$ be the point of coordinate $(x,0)$ and let $C_x$ be ...
- 401
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What’s the form of Gram matrix for right-angled hexagon
Informally, right-angled hyperbolic hexagon is a hyperbolic triangle with vertices outside infinity. I think there should be a Gram matrix for it, and what does it looks like?
(The Gram matrix here ...
- 21
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655
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How to understand the simple closed curves in torus?
Let $\alpha$ and $\beta$ be two simple closed curves in $T^2$ that intersect each other in one point.
We identify $\alpha$ with $(1,0)\in \mathbb{Z}^2$ and $\beta$ with $(0,1)\in\mathbb{Z}^2$. Let $(...
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559
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the paraboloid model for hyperbolic space [closed]
In Thurston's Three-Dimensional Geometry and Topology, he gives a recipe for a non-standard model for hyperbolic space which he calls the paraboloid model. I'd like to use the model to try out ...
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