# Questions tagged [hyperbolic-geometry]

The hyperbolic-geometry tag has no usage guidance.

863
questions

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### Number of regions created by r hyper-planes in n-dimensional space [closed]

I found this formula for calculating maximum number of regions created by r hyper-planes in n-dimensional space (n<=r)
...

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1
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102
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### Re: Vertices of hyperbolic triangle with given angles [closed]

I'm working on visualizing hyperbolic triangles given angles following a previous discussion on MathOverflow. The algorithm, as outlined in this MathOverflow answer, involves computing the side ...

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### Residual finiteness of hyperbolic 3-manifold groups

So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is:
Q1. If $M$ is an infinite-...

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### Hyperbolic three-manifolds that fiber over the circle

Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...

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### Describing the hyperbolic structure of punctured torus in terms of the period lattice

Let $T$ be a torus, $T^* = T - \{p\}$ be the complement of a point $p$. Let's fix a pair of generators $x,y\in\pi_1(T^*)$. Their images in $\pi_1(T)$ also generate, and will also be denoted by $x,y$.
...

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### For which quadratic number field, the algebraic integers are cusps for some Coxeter group?

Let $H^2=\{(x,y)\mid y>0\}$ be the hyperbolic upper-half plane.
Let $K=Q(\sqrt{d})$ be a quadratic number field, and $\mathcal{O}_K$ be the ring of algebraic integers in it.
Let $\Gamma=\Delta(p,q,...

2
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1
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### Simple curves on hyperbolic tori

In the paper "Simple curves on hyperbolic tori" by McShane and Rivin, they show that if $T$ is a hyperbolic once punctured torus, one can define a norm on the homology $H_1(T,\mathbb{Z})$ by ...

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### Guts of 3-manifolds for sutured manifolds and pared manifolds

I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me.
...

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### A formula for the cross-ratio in terms of hyperbolic data

Let $(\zeta_i) \subset \hat{\mathbb{C}}$, for $i = 1, \ldots, 4$, be $4$ distinct points on the Riemann sphere $\hat{\mathbb{C}}$.
We will use the following convention for the cross-ratio $CR$ of ...

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### Geodesic laminations on the 4-punctured sphere

Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...

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### Is there an absolute geometry that underlies spherical, Euclidean and hyperbolic geometry?

A space form is defined as a complete Riemannian manifold with constant sectional curvature. Fixing the curvature to +1, 0 & -1 and then taking the universal cover by the Killing–Hopf theorem ...

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### Volume of the Weeks manifold and of the 5.2 knot complement

Some computations show that the Weeks manifold and the 5.2 knot complement have the same trace field (which is $\mathbb{Q}[x]/(x^3-x+1)$) and the (hyperbolic) volume of the second is 3 times the ...

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### Gradient estimate of the eigenfunction of Laplacian on hyperbolic space

I am trying to understand the asymptotic behaviors of the gradient of the eigenfunction function of the Laplace-Beltrami operator on the hyperbolic plane $\mathbb{H}^2$. Specifically, my focus lies on ...

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### Determining a convex hyperbolic pentagon by all side lengths and two specified angle sums

We are trying to prove the following statement for convex hyperbolic pentagons which we believe should be true.
Consider a convex hyperbolic pentagon with sides of lengths $a, b, c, d, e$. Suppose the ...

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1
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102
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### Geodesic whose one end is at a ideal point

We know that every hyperbolic manifold $M$ is locally isometric to hyperbolic plane $\mathbb{H}_2$. Can we consider the inverse image, under the isometry, of the geodesic $\gamma$ that connects a ...

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### Does there exist a finite-volume hyperbolic Coxeter polytope with these properties?

I searched for a finite-volume, hyperbolic Coxeter polytope of dimension $n \geq 4$ with the following properties $a$ and $b$.
$a$) It has exactly one ideal vertex;
$b$) if a bounded facet and an ...

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2
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223
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### Isometric embeddings of $\Bbb H^3$

Consider the upper-half space model of hyperbolic $3$-space $\Bbb H^{+}_{3}$, the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature ...

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2
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590
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### Computing hypergeometric function at 1

I'm looking to compute
$${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr
1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$
for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...

2
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1
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318
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### Realizable geometrically finite hyperbolic 3-manifolds with prescribed conformal boundaries

By Bers' simultaneous uniformization theorem, if $\Gamma$ is a Fuchsian group, then $\operatorname{QC}(\Gamma)\cong \mathcal{T}(S)\times\mathcal{T}(\overline{S})$ where $S = \Bbb H^2/\Gamma$. In ...

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### Example of three dimensional atoroidal Poincaré duality group with some pathology

I am looking for a 3-manifold which is closed, aspherical, orientable, and atoroidal. And additionally I want to see an example that does not admit a fixed-point-free action on a simplicial tree. As a ...

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### Existence of finite 3-dimensional hyperbolic balanced geometry

Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...

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### Closed form ODE solutions for Jacobi field/eigenfunction of Laplacian on hyperbolic space

I'm trying to compute Jacobi fields of the hyperbolic disk $\mathbb{H}^m$ considered as a minimal hypersurface in $\mathbb{H}^{m+1}$ in the half model. References to literature or solutions to the ...

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1
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### Hyperbolicity and inequality for variety of general type

$\DeclareMathOperator\BMY{BMY}$Let $(X,H)$ be a smooth, complex, projective, polarized variety of dimension $n\geq3$, whose canonical bundle $K_X$ is big and nef.
Is it know whether the inequality $\...

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1
answer

121
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### Divergence functions in hyperbolic groups

Gromov hyperbolicity has many characterizations, one of them being the existence of a super-linear divergence function, see definition below.
We note that in $\mathbb{R}^2$ there is no divergence ...

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1
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346
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### Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?

$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$
I have a reference request for a proof for the following statement in the title:
The Teichmüller space $T_g$ of the surface $S_g$ of genus ...

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### Theorems that hold in Euclidean and Elliptic Geometry but not in Hyperbolic (and vice versa)

Question: Which theorems (if any) of plane Euclidean geometry continue to hold in Elliptic geometry but don't hold in Hyperbolic? And are there theorems that are valid in Euclidean and Hyperbolic ...

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113
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### Expansion of metric near boundary of 3 dimensional Poincaré-Einstein/hyperbolic manifolds

In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as
$$g = \frac{...

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1
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### Number of ergodic transverse measures for geodesic laminations - bounded by the genus?

Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...

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### Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces

I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...

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### Asymmetric minimal surfaces in $H^3$

Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by
$$y^2 ...

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1
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### Solving equations in hyperbolic groups and subgroups of isometry of a Gromov hyperbolic space

Let $\Gamma$ be a hyperbolic group. Let $g$, $\gamma\in \Gamma$ freely generate a non-abelian semigroup (in particular, they don't commute and have infinite order). Does the equation $g\gamma^n=h^m$ ...

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546
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### Is every uniform hyperbolic linear space infinite?

I start with definitions.
Definition 1. A linear space is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ satisfying three axioms:
(L1) for any distinct ...

3
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1
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### Group action of $\text{SL}(2, \mathbb{C})$

Let's denote upper-half space model of hyperbolic 3-space identified with quaternions as $$\text{H}^{+}_3 = \biggr\{z+wj\in \mathbb{C}\oplus \mathbb{R}^{+}j\biggr\}\tag{1}$$
Then a group action $\rho :...

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1
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### Upper-half space model of $\text{H}_3$

Does $SL(2,\mathbb{C})\times SL(2,\mathbb{C})$ have a natural (unitary) left-action on $\mathcal{L}^2(\text{H}^{+}_{3})$?
If $G$ is a unimodular Lie group and $K$ is a compact subgroup, $G\times G$ ...

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2
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### Why is a plane graph Delaunay realizable if stellating a face makes the graph inscribable?

I have come across the following lemma in several papers (for instance see lemma 2.2 in this) (and some authors state this the proof immediately follows from basic properties of the stereographic ...

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### Which pairs of conjugates of $\left(\begin{smallmatrix} 1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$ generate $\operatorname{SL}(2,\mathbb{Z})$?

When do two distinct conjugates of $U := \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ generate $\DeclareMathOperator\SL{SL}\SL(2,\mathbb{Z})$? The classic example is $U,L^{-1}$, where $L = \...

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### Relation of geometric and polyhedral convergence

By Proposition 3.10(i) of Jorgensen and Marden's 1990 Algebraic and geometric convergence of Kleinian groups, "[A] sequence $\{G_n\}$ of Kleinian groups converges geometrically to a Kleinian ...

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### Minimum diameter of spherically-inverted topological balls

Let $U$ be the closed unit ball in $\mathbb{R}^3$. Let $S$ be a round sphere whose center is in $U$ with radius at least $\delta_1 > 0$. Suppose $B$ is a closed topological ball of Euclidean ...

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### Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul

In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...

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### Markov property for groups?

My question again refers to the following article:
Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:...

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### Why can't a Lie group act transitively on a finite volume hyperbolic manifold?

In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...

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### Geodesics in free homotopy classes and the fundamental group

Let $\mathcal{H}$ be the upper half-plane and $\Gamma$ be a cocompact, torsion-free Fuchsian group. The quotient space $X=\Gamma\backslash \mathcal{H}$ is a smooth closed Riemann surface and there is ...

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### Hyperbolic or spherical analogue to the quadrilateral inequality

This is a reference request.
Let $x, y, z, w \in \mathbb{R}^n$. Then we have a so-called "quadrilateral inequality":
$$
0 \leq \lVert x-y-z+w \rVert^2 = \lVert x-y\rVert^2 + \lVert z-w \...

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0
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75
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### Understanding logarithmic law for geodesics

I was reading this seminal paper
https://projecteuclid.org/journals/acta-mathematica/volume-149/issue-none/Disjoint-spheres-approximation-by-imaginary-quadratic-numbers-and-the-logarithm/10.1007/...

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### Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?

I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...

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### Monotonicity of root of hyperbolic function

For $\kappa \geq \alpha>0$ and $y \geq 0$, consider the following equation:
$$\sqrt{1-\frac{\alpha }{\kappa }} \tanh \left(y \sqrt{1-\frac{\alpha }{\kappa }}\right)=\tanh \left(y-\frac{\alpha }{2}\...

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### Is the canonical map from isometry group of a Gromov hyperbolic space to homeomorphisms of its Gromov boundary injective?

Suppose X is a proper Gromov hyperbolic space and $\partial X$ is its Gromov boundary. It is well-known that there is a canonical group homomorphism $\Phi$ from the isometry group of X to the group ...

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### The relationship between convex hulls

Consider a (f.g., classical) Schottky group acting on $\mathbb H^3$; consider a convex hull of the limit set $C(\Lambda)$ and a convex hull of a closure of an orbit of a point on $\mathbb CP^1,$ $C(\...

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### Understanding $\kappa$-cones

I recently came across the concept of a $\kappa$-cones of a metric space (Chapter I.5.2) of Bridson and Haefliger's book. In their Proposition 5.8, the provide some intuition of $\kappa$-cones by ...

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2
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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 $M$ ...