Questions tagged [hyperbolic-geometry]

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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 ...
Pedro's user avatar
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2 votes
0 answers
184 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) ...
Amirhossein's user avatar
8 votes
1 answer
634 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) ...
Agelos's user avatar
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1 vote
1 answer
146 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+...
Tito Piezas III's user avatar
2 votes
0 answers
112 views

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 $\pi(K) = \pi_1(S^3 \setminus K)$...
Calvin McPhail-Snyder's user avatar
1 vote
0 answers
57 views

Cohomology of cocompact lattices in hyperbolic spaces

I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...
TSU's user avatar
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1 vote
1 answer
108 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 ...
Nandakumar R's user avatar
  • 4,577
2 votes
0 answers
46 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 ...
Quanta's user avatar
  • 31
4 votes
2 answers
195 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 ...
Julian Chaidez's user avatar
2 votes
0 answers
49 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\...
taylor's user avatar
  • 385
4 votes
1 answer
104 views

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'...
Ethan Dlugie's user avatar
  • 1,045
2 votes
0 answers
58 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 ...
Jacques's user avatar
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1 answer
85 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 ...
DavidHume's user avatar
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0 answers
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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 \...
HERMIT_WELL's user avatar
2 votes
0 answers
56 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: $...
Narasimham's user avatar
2 votes
0 answers
153 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 ...
RegularGraph's user avatar
3 votes
0 answers
59 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 ...
Calvin McPhail-Snyder's user avatar
6 votes
1 answer
113 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 $(X,d)$. ...
Mark Hagen's user avatar
1 vote
0 answers
36 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 ...
Nandakumar R's user avatar
  • 4,577
5 votes
0 answers
145 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(\...
Atlas Tasilli's user avatar
2 votes
0 answers
89 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 ...
user avatar
8 votes
0 answers
87 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\...
Taras Banakh's user avatar
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0 votes
1 answer
117 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 ...
Taras Banakh's user avatar
  • 38.7k
0 votes
1 answer
72 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 ...
Math_Newbie's user avatar
5 votes
0 answers
126 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 ...
Random's user avatar
  • 793
5 votes
1 answer
246 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 ...
Adrien B's user avatar
3 votes
0 answers
85 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?
Daniel Sebald's user avatar
1 vote
1 answer
156 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/...
John Depp's user avatar
  • 147
3 votes
1 answer
55 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 ...
Taras Banakh's user avatar
  • 38.7k
0 votes
0 answers
60 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 ...
KAK's user avatar
  • 179
0 votes
0 answers
75 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}^...
AMHG's user avatar
  • 23
11 votes
2 answers
1k views

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?...
David.D's user avatar
  • 423
1 vote
0 answers
78 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 ...
user6419's user avatar
  • 411
5 votes
3 answers
185 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 ...
Calvin McPhail-Snyder's user avatar
3 votes
0 answers
264 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,...
KAK's user avatar
  • 179
5 votes
1 answer
183 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 $3$...
user126154's user avatar
1 vote
0 answers
132 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 $(x,y\cosh(r))$...
lille nordmann's user avatar
0 votes
0 answers
38 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 ...
apg's user avatar
  • 532
1 vote
2 answers
187 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. ...
gaoqiang's user avatar
  • 119
8 votes
1 answer
562 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 ...
Anubhav Mukherjee's user avatar
1 vote
1 answer
163 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 ...
lille nordmann's user avatar
0 votes
0 answers
49 views

A question about quasi hyperbolic metric

I am reading "Quasiconformally homogeneous domains" by F. W. Gehring and B. P. Palka. Let $D$ be a proper subdomain of $\mathbb{R}^n.$ Define the function $\rho.(x) = \frac{1}{\mathit{dist}(...
dip's user avatar
  • 11
3 votes
1 answer
148 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 ...
Colby's user avatar
  • 33
1 vote
0 answers
68 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 ...
Melanka's user avatar
  • 549
0 votes
1 answer
93 views

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 $l_1$ ...
KAK's user avatar
  • 179
2 votes
1 answer
88 views

Pair of laminations that fill on a closed surface

Let $S$ be a hyperbolic surface of genus $g \geq 2$. A discrete geodesic lamination on $S$ is a set of disjoint, simple, closed geodesics. Let $L_{1}$ and $L_{2}$ be two discrete geodesic laminations ...
AMHG's user avatar
  • 23
3 votes
1 answer
82 views

A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it has zero geometric intersection numbers with $\Gamma$

Let $\Sigma$ be a compact oriented connected bordered surface other than the pair of pants. Let $\Gamma:=\{\gamma_i\}$ be a finite collection of simple closed curves on $\Sigma$ such that each ...
Random's user avatar
  • 793
4 votes
1 answer
153 views

Infinitely divisible elements in Gromov hyperbolic groups

An element $g\in G$ in a group $G$ is called infinitely divisible if $b=y^n$ for infinitely many different $n\in {\Bbb Z}$. It is not hard to find a finite CW-complex (or even a compact manifold) ...
Misha Verbitsky's user avatar
1 vote
0 answers
55 views

Regarding equality of angles of a hyperbolic polygon [closed]

Let $P$ be a hyperbolic $4n$ gon having opposite sides with equal length and the sum of all interior angles is $2\pi$. Then is true that opposite angles are equal also?
KAK's user avatar
  • 179
0 votes
0 answers
35 views

For fixed $B > 0$, asymptotically describe the upper half plane complex zeros of $B\cos(\omega z) = iz$ as $\omega \to \infty$

Long story short, I am seeking to evaluate an oscillatory integral via complex methods, whose partial fraction expansion has $B\cos(\omega z) - iz$ in the denominator, where $B, \omega$ are positive ...
maxematician's user avatar

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