Questions tagged [hyperbolic-geometry]
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864
questions
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Fibration of hyperbolic 3-manifold
A fibration of a manifold $\phi: M \to S^1$ gives rise to a short exact sequence
$$
1 \to \pi_1(N) \to \pi_1(M) = \mathbb{Z} \overset{f_\ast}{\to} 1
$$
where $N$ is the fiber.
I've heard that, if $M$ ...
0
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0
answers
64
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Are Gromov-hyperbolic groups roughly starlike? [duplicate]
Given a Cayley graph of a finitely generated Gromov-hyperbolic group $G$, does there exists $R>0$ such that every element $g \in G$ is at most distance $R$ away from a geodesic ray starting at ...
0
votes
0
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13
views
Relation between the "s" parameter of Ungar's theory of hyperbolic geometry and the eccentricty in the 2D case
In Ungar's theory of hyperbolic geometry for the Minkowski model, there is a parameter $s>0$ which controls the curvature of the hyperbolic segments:
Ungar's theory is not very well-known. An ...
13
votes
0
answers
195
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Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?
$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
4
votes
1
answer
188
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Conformal map between flat and hyperbolic torus with a boundary
I am confused because I can define two very different complex structures on the torus with a puncture/boundary.
For my first construction, I can imagine removing a disk from a flat torus, inheriting ...
5
votes
1
answer
180
views
Can hyperbolic surfaces approximate every connected compact metric space?
Let $X$ be a connected compact metric space.
Question: Is there a sequence of compact hyperbolic surfaces (the curvature may differ between surfaces) that converges to $X$ in the Gromov-Hausdorff ...
6
votes
1
answer
451
views
Examples of groups that are unknown to be acylindrically hyperbolic
Let $G$ be a group. We say that $G$ is acylindrically hyperbolic (for short, AH) if $G$ admits an isometric, acylindrical, and non-elementary action on some Gromov hyperbolic space $X$.
Here is the ...
5
votes
1
answer
226
views
Cancellation of elements in the Gromov boundary of a free group
Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
1
vote
0
answers
183
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Simple left earthquakes are dense
i´ve been studying an article from W. P. Thurston about hyperbolic geometry, there, he defines something called left earthquake, whose definition is as follows:
Definition. If $\lambda$ is a geodesic ...
2
votes
0
answers
229
views
A Question about an article by Birman, Series
Birman and Series in their article GEODESICS WITH BOUNDED INTERSECTION NUMBER ON
SURFACES ARE SPARSELY DISTRIBUTED proved that the set of points on a hyperbolic surface (possibly with boundary) ...
8
votes
1
answer
822
views
Problem 3.14 from Kirby's list
In his famous list of Problems in Low-Dimensional Topology, Kirby states the following as Problem 3.14 (B), which is attributed to Thurston:
Conjecture: Suppose $G$ (an arbitrary group I suppose) ...
1
vote
1
answer
165
views
Bring's curve $\sum_{i=1}^5 x_i^k = 0$ for $k = 1,2,3$ and an analogue $\sum_{i=1}^6 y_i^k = 0$ for $k = 1,2,4,7$
Bring's curve or Bring's surface with genus 4 and $5!=120$ automorphisms can be given by the homogeneous equations,
$$x_1+x_2+x_3+x_4+x_5 = x_1^2+x_2^2+x_3^2+x_4^2+x_5^2 = \\x_1^3+x_2^3+x_3^3+x_4^3+...
2
votes
0
answers
141
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Can distinct meridians commute in a knot group?
Suppose I have a knot $K$ in $S^3$. Given a diagram $D$ of $K$ I get the Wirtinger presentation $\langle x_1, \dots, x_a \mid r_1, \dots, r_c\rangle$ of its knot group $\pi(K) = \pi_1(S^3 \setminus K)$...
1
vote
1
answer
148
views
Tiling the hyperbolic plane by non-regular quadrilaterals
We add a bit to Which polygons tessellate the hyperbolic plane?.
Question: Are there hyperbolic quadrilaterals with all angles different (not necessarily irrational fractions of π) that tile the ...
2
votes
0
answers
58
views
Examples of elementary group of isometries of the ideal boundary of hyperbolic plane
A Riemannian manifold $(X,g)$ is called Hadamard manifold if it is complete and simply connected and has everywhere non-positive sectional curvature. An example of such a manifold would be the ...
5
votes
2
answers
253
views
Canonically representing the monodromy of a hyperbolic manifold fibered over $S^1$
Let $Y$ be a hyperbolic manifold that fibers over $S^1$, with fibration $\pi:Y \to S^1$ with fiber $\Sigma$. Thurston states that the monodromy $\phi:\Sigma \to
\Sigma$ of this projection is then ...
2
votes
0
answers
109
views
Double quotient integral formula on $\Gamma \backslash G /K$
Let $G=\text{SL}(n,\mathbb R)$, $\Gamma=\text{SL}(n,\mathbb Z)$ and $K=\text{SO}(n,\mathbb R)$. Consider the double coset space $X= \Gamma \backslash G /K$ and its fundamental domain $\mathcal F\...
4
votes
1
answer
119
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Inheritance of arithmeticity properties in orbifold strata
Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...
2
votes
0
answers
67
views
Maximal orders and surface subgroups of even genus
Let $A$ be a quaternion algebra over a totally real number field $k$. Suppose that $A$ splits at exactly one real place of $A$. Let $\mathcal{O}$ be a maximal order in $A$. Then $\mathcal{O}$ contains ...
1
vote
1
answer
143
views
$1$-Lipschitz map from hyperbolic to Euclidean plane
I'm trying to find a reference to the following statement.
Define a function $f$ from the hyperbolic plane (in the Poincaré unit disc model using polar coordinates) to the Euclidean plane (using polar ...
0
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0
answers
102
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The upper bound of hyperbolic cosine function in complex plane
I want to find the upper bound of the following function :
\begin{equation}
M(\lambda)=\max _{E \in \mathbb{C}}\left|L_{+}-L_{-}\right|
\end{equation}
where
\begin{equation}
\begin{aligned}
& 4 \...
2
votes
0
answers
63
views
Mohr circle curvatures in constant negative Gauss curvature K Chebyshev net
In order to verify vanishing normal curvature $\kappa_n$ everywhere for asymptotic lines as hyperbolic geodesic representation of a Chebyshev net I have used relations represented in the Mohr circle:
$...
2
votes
0
answers
173
views
Representation determined by traces
A discrete, faithful representation of a surface group $G:\pi_1(S_g) \to PSL_2(\mathbb{R})$ is determined, up to conjugacy by $PGL_2(\mathbb R)$, among such representations by the squares of traces of ...
3
votes
0
answers
67
views
Discreteness of volumes of boundary-parabolic representations
Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
6
votes
1
answer
142
views
Translation length on annular curve graphs
Question about curve stabilisers acting on annular curve graphs, plus context since I'm interested in being fact-checked.
Definition: let the group $G$ act by isometries on a metric space $(X,d)$. ...
1
vote
0
answers
37
views
Tiling the hyperbolic plane with mutually-non congruent equal area triangles
This post continues On tiling the plane with non-congruent, equal area triangles with each edge having a unique length
Can the hyperbolic plane be tiled by pair-wise non-congruent equal area ...
5
votes
0
answers
159
views
Intersection of orbits of earthquake flow on Teichmüller space
Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
2
votes
0
answers
104
views
Further directions in representations of surface group into a Lie group
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$I studied the interpretation of Teichmuller space as a representation space for surface groups in $\PSL(2,\mathbb{R})$.
Now I am planning to ...
8
votes
0
answers
93
views
Is the hypotenuse operation associative in every Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
0
votes
1
answer
135
views
Is a right triangle with the given cathetus and the opposite angle constructible in the absolute plane?
Question. Is it possible to construct a right triangle with a given cathetus $a$ and a given opposite angle $\alpha$ using only compass and ruler in the absolute geometry (so without the axiom of ...
0
votes
1
answer
87
views
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 ...
6
votes
0
answers
146
views
What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?
Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
5
votes
1
answer
417
views
Ricci flow negative curvature
We denote by $\mathbb{H}^n$ the hyperbolic plane of dimension $n$.
I don't know much about the Ricci flow, so my first question is probably naïve : can one define a normalized Ricci flow such that the ...
4
votes
0
answers
93
views
Is there a 5-cell-600-cell honeycomb?
Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
1
vote
1
answer
176
views
A question from the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel
In the paper "Hyperbolic rigidity of higher rank lattices" by Thomas Haettel, the author has made the following statement in the proof of Corollary D in page no. 18 ( https://arxiv.org/pdf/...
3
votes
1
answer
64
views
Comparing the areas of polygons via equidecomposability in the hyperbolic plane
It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles.
Question. Is an ...
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votes
0
answers
76
views
Number of tiles inside a region of a hyperbolic tiling
Let $\mathbb{D}$ be the Poincare disc model of hyperbolic geometry with $\{p,q\}$ tiling on it.
In this, paper authors calculated the number of tiles on any circular region centered at a vertex or ...
0
votes
0
answers
80
views
Deformation of hyperbolic structures
Let M be a hyperbolic 3-manifold, let $g_{t}$ be a smooth family of hyperbolic metrics on M. It is known that in this case we can find a family of devlopping maps $dev_{g_{t}}:\tilde{M} \to \mathbb{H}^...
11
votes
2
answers
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Is there a contractible hyperbolic 3-orbifold of finite volume?
Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...
1
vote
0
answers
87
views
Almost modularity of Belyi curves and etale fundamental group of non-Belyi curves
Belyi's theorem states that every curve defined over $\mathbb{\bar Q}$ is almost modular (obtained from $\mathbb{H}^2/\Gamma,\ \Gamma$ a finite index subgroup of $PSL(2,\mathbb{Z})$), after ...
5
votes
3
answers
224
views
Ideal triangulations of $3$-manifolds with "cusps" of genus $\ge 2$
Typically when one thinks about ideal triangulations of a $3$-manifold the link of each ideal vertex is a circle, so the ideal points correspond to toroidal cusps; alternatively, one can truncate the ...
3
votes
0
answers
335
views
Representations of triangle groups
$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up.
Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
5
votes
1
answer
241
views
Question about and good reference for Kahn and Markovic result
As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:
Let $M$ be a compact hyperbolic $3$...
1
vote
0
answers
201
views
Hyperbolic vs Euclidean balls [closed]
I'm trying to prove that, in the Poincaré half-space of dimension 2, a hyperbolic ball with center $P:=(x,y)$ and radius $r$ is exactly, as a set of points, a euclidean ball with center $(x,y\cosh(r))$...
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votes
0
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43
views
Hyperbolic random geometric graphs with less clustering
The hyperbolic random geometric graph $G_{\mathbb{H}}$ consists of a $N$ uniformly random points (within a disk of radius $R$ centred at the origin) of the hyperbolic plane $\mathbb{H}$, connected ...
2
votes
2
answers
259
views
Is a local isometry of the hyperbolic plane the restriction of a global isometry?
The origin question: Let $\Omega \subset \mathbb{H}^2$ be a domain of the hyperbolic plane $\mathbb{H}^2$. Let $u: \Omega \to \mathbb{H}^2$ be injective and an isometry from $\Omega$ to its image. ...
8
votes
1
answer
617
views
On trivial mapping class group of 3-manifolds
What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
1
vote
1
answer
322
views
How to prove that the hyperbolic space is $\delta$-hyperbolic
How can I prove that the hyperbolic space $\mathbb{H}^n$ (in any of its realization as hyperboloid, Poincaré disc or Poincaré half-space) is $\delta$-hyperbolic? For the moment I am not interested in ...
3
votes
1
answer
195
views
Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?
It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed ...
1
vote
0
answers
74
views
Translate of a geodesic that goes through a fixed point on $\mathbb{H}$
Consider the complex upper half plane $\mathbb{H}$ with the hyperbolic geometry. Fix a point $z \in \mathbb{H}$ and also a geodesic $c$. I want to find a hyperbolic translation $\gamma c$ passes that ...