# Questions tagged [hyperbolic-geometry]

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature. For questions on hyperbolas in planar geometry, use the tag conic-sections.

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### How is the real line a geodesic in the complex upper half-plane?

I'm in the middle of trying to prove that all hyperbolic isometries are Möbius transformations in the upper half-plane and I keep seeing people mention looking at points on the real line and ...
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### Hyperbolic surfaces with only one short geodesic

$\textbf{Question}$: Let $R>0$. Does there exist a compact hyperbolic surface $S$ which has one and $\underline{only\ one}$ primitive geodesic of length $\le R$? I am aware of the fact that the ...
1 vote
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### Obtaining the Area Formula of a Fundamental Domain from the Gauss-Bonnet Theorem

First of all, sorrying for posting a similar question again but this is the improved version (See reference section). A special case of Theorem 10.6.4 of Beardon's ...
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### Derivation of a Fundamental Domain's Area Formula from the Gauss-Bonnet Theorem [closed]

A special case of Theorem 10.6.4 of Beardon's book about possible shapes of Dirichlet domains have been discussed in the following result from Yiltekin-Karatas's dissertation. Lemma 1 (Page.9) ...
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### Origin of a Relation in the Proof of Theorem 10.6.4 in Beardon's Book

I'm studying the proof of the following theorem in Beardon's book. Theorem 10.6.4: A group $G$ is a $(p, q, r)$-Triangle group if and only if it is a discrete group of the first kind with signature ...
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1 vote
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### How do I translate the description of geodesics of hyperbolic space in the hyperboloid model to the Poincaré ball and half-space models?

I have a nice description of the geodesics of hyperbolic space in the hyperboloid model as intersections of $2$-planes with the hyperboloid, as given in this answer. As discussed there, if such a ...
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### Example of an Accidental Cycle in a Hyperbolic Polygon

I am trying to understand the concept of accidental cycles of a hyperbolic polygon through a numerical example. Definition: Accidental cycle If an elliptic cycle transformation is the identity then we ...
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### Understanding Cycles in Hyperbolic Geometry: Definitions and Examples

I'm having trouble understanding the concept of cycles in the hyperbolic geometry, in particular, in the following works. Definitions Definition 17. (Cycle) of ...
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### Understanding the Projection of a Subset $H$ as a Cylinder in the Context of Fuchsian Groups

I'm currently studying Beardon's book and I'm having trouble understanding a passage in Section 10.4 regarding the signature of a Fuchsian group. Beardon mentions that §10.4. The Signature of a ...
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1 vote
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### Understanding the Absence of Odd-sided Shapes in Dirichlet Domains: Theorems and Discussion

I am studying the shape of dirichlet domains. I understood that the following Theorem 10.5.1 of Beardon provides the bounds on the number of sides. And the Umemoto's paper (See References) also ...
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### Computations of tensor on a Riemannian Manifold.

I am trying to get some practice in computing tensors in Riemannian and Pseudo-Riemannian manifolds. Is there somewhere that I can check my results? For example right now I am computing the ...
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### Geodesics in Hyperbolic Disk

Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{D}^2/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{D}^2$...
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### Paths in the hyperbolic disk

Let $\Gamma \le \operatorname{PSL}(2,\mathbb{R})$ be a discrete subgroup and $\Sigma:=\mathbb{D}^2/\Gamma$ be the quotient. Then $\Sigma$ is a hyperbolic surface whose universal cover is $\mathbb{D}^2$...
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### Hyperbolic Regular Polygons: Construction and Visualization

I am working on visualising a polygon in a hyperbolic space using a HyperbolicRegularPolygon class in SageMath, which constructs regular polygons in the hyperbolic ...
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### Hyperbolic trigonometry NOT in Poincare disk

There are a lot of hyperbolic trigonometric identities derived in Poincare disk model that resemble similar identities from the Euclidean geometry. For example, the analogue of the Pythagoras theorem ...
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### Understanding Assumed Angles in Umemoto's Hexagon Subdivision

In Umemoto's thesis 1 on Dirichlet fundamental domains for Fuchsian groups, Theorem 24 involves assumed angles for subdivided hexagons in the proof (Fig.18 on page 35). However, the rationale behind ...
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### Exploring Polygon Drawing in Hyperbolic Space with Integer Factor Angles: Seeking Simpler or Standard Methods

I am currently attempting to draw polygons in hyperbolic space with angles that are integer multiples of pi, i.e., $\pi / k$ for k an integer. I am particularly interested in determining the vertex ...
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### Hyperbolic Reflection of polygon

I'm working on visualizing the reflections of a polygon in the Poincaré disk along each side of it using SageMath. The figures below show the reflections of a polygon (a 4-gon and a 3-gon, the ...
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### Bipartite intersection graph

Recently, I'm trying to learn geodesic currents on free groups through a paper by Kapovich and Lustig. Let $\langle, \rangle: \overline{CV}(F_N)\times Curr(F_N)\to\mathbb{R}$ be the intersection form ...
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### Why don't I see any mention of fundamental domain in the context of non-discrete Lie Group acting on Riemannian manifolds

Let $G$ be a non-discrete Lie Group acting properly on a Riemannian manifold $M$ by its isometries. When $G$ is discrete, there are plenty of literature on Fundamendal Domains (FD), but a quick ...
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### What is the formula for the metric of an N-hyperbolic hyperboloid?

I saw this question on MSE. It's about the general formula of the metric of an N-sphere. I was wondering, is there a similar general formula for the metric of an N-hyperbolic (simply connected, one-...
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1 vote
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### $\Gamma <\mathrm{PSL}_2(\mathbb{R})$: non-compact if contains parabolic element.

It seemingly is a fact that a discrete subgroup of $\mathrm{PSL}_2(\mathbb{R})$ acting on hyperbolic space cannot be compact if it contains a parabolic element. I was wondering if the following proof ...
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### How is the Poincare disc model or the upper half plane model a representation of hyperbolic space?

I am currently working on these two models and I don't understand the connection between them and hyperbolic space. In case of spherical geometry one can imagine everything well as a 2 dimensional ...
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### Visualisation of isotopy

How can I visualise the meaning of isotopy that appears while defining Teichmuller space? Can you suggest a picture where two maps are not isotopic? I want more clarification about isotopyic maps. ...
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### What does "Motions along translation axes induce the opposite orientations on the region they bound" mean?

I am reading a paper and they mention the following: "Two isometries of the hyperbolic plane are said to be co-parallel if they have disjoint axes and the motions along these axes induce the ...
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### Realizable subsets of the sphere at infinity

Suppose we have an arbitrary finite collection $X = (x_i)_{i = 1} ^ n$ of the boundary sphere $\partial D$ where $D$ represents the Poincare disk. Does there exist a (discrete?) subgroup $G$ of the ...
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### What is the abelianization of $\operatorname{Aut}(F_2)$?

Let $F_2$ be the free group of rank 2. What is the abelianization of $\operatorname{Aut}(F_2)$? There is a surjection $\operatorname{Aut}(F_2) \rightarrow\operatorname{ GL}_2(\mathbb{Z})$, so we at ...
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### Seeking References for Visual Verification of Dirichlet Regions in Triangle Groups

I'm self-studying the impact of base point placement on the shape of Dirichlet regions within triangle groups. Theoretical findings suggest the region is a quadrilateral when the base point lies on a ...
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### Every Hyperbolic manifold is a quotient of a Hyperbolic space by a certain discrete group

I started reading about Hyperbolic manifolds here: https://en.m.wikipedia.org/wiki/Hyperbolic_manifold and I didn't understand the following paragraph in the first section of Rigourous definition: ...
When working with subgroup of Klenian groups (or in general just asking for discrete subgroup forgetting about the dimension of Hyperbolic space) the classic definition of limit set $\Lambda(\Gamma)$ ...