# 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 ...

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### 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) ...

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### 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) ...

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### 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+...

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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 $\pi(K) = \pi_1(S^3 \setminus K)$...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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 ...

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### 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\...

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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'...

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### 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 ...

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### $1$-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 ...

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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 \...

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### 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:
$...

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### 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 ...

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### 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 ...

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### 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 $(X,d)$. ...

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### 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 ...

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### 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(\...

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### 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 ...

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### 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\...

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### 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 ...

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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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### 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 ...

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### 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 ...

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### 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?

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### 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/...

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### 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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### 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 ...

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### 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}^...

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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?...

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### 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 ...

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### 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 ...

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### 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,...

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### 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 $3$...

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### 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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### 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 ...

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### 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. ...

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### 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 ...

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### 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 ...

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### 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}(...

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### 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 ...

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### 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 ...

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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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### 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 ...

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### 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 ...

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### 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) ...

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### 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?

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### 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 ...