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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How do I prove the symmetry of the Hyperbolic metric on the Half-Plane?

Okay so the preamble I'm working with is that in the upper half-plane model of hyperbolic space, we are equipped with an inner product defined on a vector $v$ originating at the point $z=x+iy$ as: $\...
sympie's user avatar
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Pair of pants with k-genus

Let $\mathbb{H}$ be the hyperbolic plane. An usual($0$-genus) pair of pants $P_0$ is a $2$ dimensional sphere with $3$ holes and constant curvature $-1$. The boundary of a pair of pants consists of ...
Quanta's user avatar
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Verifying that $M$ has curvature $-1$ iff $f''=f$ when $g=dr^2+f(r)^2d\theta^2,M=[0,\infty)_r\times S_\theta^1$

I am currently at a difficult position, because I have to check some definitions/examples regarding hyperbolic surfaces, but I have not taken a proper course on Riemannian manifolds or surfaces in the ...
Epsilon Away's user avatar
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Congruent vertices are equidistance from the center of a Dirichlet Domain for a Fuchsian group

Let $D_x(G)$ be a Dirichlet domain for a Fuchsian group $G$, where x is the center of the Dirichlet domain then show that $d(x,z_1)=d(x,z_2) $ where $z_1,z_2$ are congruent vertices of the Dirichlet ...
jay sri krishna's user avatar
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Euclidean Diameter of $G-$ translates of Fundamental domain goes to $0.$ for a Fuchsian group $G$

Lemma : Let $G=\{Id, g_1,g_2,...\}$ be a Fuchsian group acting on $\mathbb{U}$ , where $\mathbb{U}$ is the upper half plane and $F$ be locally finite fundamental region for $G.$ Then the Euclidean ...
jay sri krishna's user avatar
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Symmetric hyperbolic 3-manifold

Suppose one has a complete closed hyperbolic 3-manifold upon which there is an anti-conformal involution (a symmetry). Can one represent the manifold as a quotient of $\mathbb{H}^3$ by some discrete ...
Alex's user avatar
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How do you *accurately* calculate distances in the hyperboloid model of hyperbolic geometry?

In the hyperboloid model, the distance between two points has a straightforward calculation: $D(\mathbf{u}, \mathbf{v}) = \operatorname{arcosh}(-B(\mathbf{u}, \mathbf{v})) = \operatorname{arcosh}(\...
D0SBoots's user avatar
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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 ...
Lille Nordmann's user avatar
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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 ...
Rowing0914's user avatar
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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) ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
S_d_pap's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
serpens's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
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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 ...
Rowing0914's user avatar
2 votes
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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 ...
Learning Math's user avatar
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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-...
Il Guercio's user avatar
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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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Riemannian geometry and Symmetries, Hyperbolic Spaces

I am currently trying to connect Hyperbolic geometry through several models with Riemannian geometry. At first I have transformed the metric tensor from sphere in $R^3$ and the pseudo-sphere in $R^{2,...
S_d_pap's user avatar
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1 answer
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Action of $SL_2(\mathbb{Z})$ on the projective plane over $\mathbb{Z}_p$

The group $SL_2(\mathbb{Z})$ act on the projective spaces $P(\mathbb{Z}_p)$ and the upper half of the complex plane $\mathbb{H}$ by linear fractional transformations. I am wondering whether there is a ...
QMath's user avatar
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What is the intuition behind the classification of möbius transformation in hyperbolic, elliptic and parabolic transformations?

I am currently reading through Katok on fuchsian groups chapter 2.1. There the author introduces a classification of PSL(2, R) by the trace of the tranformation. It then says that hyperbolas are in ...
struggling_student's user avatar
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Constructing hyperbolic triangles with specific angles

I’m trying to investigate hyperbolic tessellations and how to construct them. After doing lots of reading I’ve found that given a hyperbolic triangle with angles $\frac{\pi}{m}, \frac{\pi}{n}, \frac{\...
Gracie's user avatar
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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 ...
struggling_student's user avatar
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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. ...
Subash Chandra Behera's user avatar
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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 ...
USer12323123's user avatar
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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 ...
discretephenom's user avatar
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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 ...
stupid_question_bot's user avatar
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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 ...
Rowing0914's user avatar
1 vote
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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: ...
user32415's user avatar
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Limit set of subgroup of Hyperbolic Isometry

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 $\Lambda(\Gamma)$ ...
Augusto Matteini's user avatar
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Prove distance formula in polar coordinates [closed]

I have seen this equality in some youtube video on hyperbolic geometry, but I want to understand the proof of it. dist$((r_1,\theta_1),(r_2,\theta_2)) = \text{arcosh}(\text{cosh}r_1\text{cosh}r_2 - \...
Just do it's user avatar
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1 answer
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Horosphere of hyperbolic space in Minkowski space

In their paper about decomposition of cusped hyperbolic manifold Epstein and Penner state that given a point $x\in L$ (where with $L$ they denote the positive component of the light-cone) that the ...
Augusto Matteini's user avatar
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Curvature in hyperbolic manifolds

I was wondering that, if I have a manifold $M=\mathbb{H}^n / \Gamma$ (where $\Gamma$ is a discrete group of isometries of the hyperbolic space), how would the curvature elements of this manifold ...
Rafael Carrasco's user avatar
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Can a Lie group act transitively on a geometrically finite hyperbolic manifold?

Let $ M $ be a hyperbolic manifold. If $ M $ has finite volume (this includes all compact hyperbolic manifolds of course) then no Lie group can act transitively on $ M $. But what about geometrically ...
Ian Gershon Teixeira's user avatar
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Hyperbolic isometry and product

Let $g:D^n\rightarrow D^n$ be a hyperbolic isometry, so it is a mobius transformation. Then, $g$ can be expressed as a composition $T\circ R$ where $T$ is a translation such that $T(0)=g(0)$ and $R$ ...
monoidaltransform's user avatar
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colinearity between complex vectors

You can find if two rea-valued vectors $x$ and $y$ in $\Re^n$ are colinear by calculating. $\cos(\theta) = \frac{\langle x, y \rangle}{\lvert x\rvert\,\lvert y\rvert} $ For complex vectors ($z \in C^n$...
user3284182's user avatar
1 vote
1 answer
149 views

A discrete and faithful higher genus closed surface group representation $\Gamma \to \mathrm{PSL}(2, \mathbb{C})$ with at least one parabolic value?

Let $g \ge 2$ and $\Gamma := \langle a_1, b_1, \ldots, a_g, b_g\ |\ [a_1, b_1] \cdots [a_g, b_g] \rangle$, and let $\rho : \Gamma \to \mathrm{PSL}(2, \mathbb{C})$ denote a representation which is ...
Geoffrey Sangston's user avatar
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Quotient of an ideal triangle in the upper half-plane by a cyclic group.

Let $T$ be an ideal triangle in the Poincare upper half-plane $\mathbb{H}$ with the point $i$ as its "circumcenter" (by which I mean that the point $i$ is the center of symmetry of this ...
kodyv's user avatar
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Find the area of a geodesic triangle in the hyperbolic space

The hyperbolic space of dimension 2 can be modeled via the upper half plane $H^2 = \{(u, v) \in \mathbb{R^2} | v > 0\}$ equipped with the metric $g(u, v) = \frac{1}{v^2} I_2$, with $I_2$ the ...
user996159's user avatar

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