Questions tagged [hyperbolic-geometry]
The hyperbolic-geometry tag has no usage guidance.
861
questions
4
votes
1
answer
334
views
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 ...
1
vote
0
answers
81
views
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 $x,y$.
...
4
votes
1
answer
170
views
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
votes
1
answer
88
views
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 ...
-3
votes
0
answers
43
views
Visualisation of isotopy [migrated]
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. ...
2
votes
1
answer
146
views
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.
...
2
votes
1
answer
131
views
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 ...
3
votes
1
answer
87
views
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 ...
10
votes
3
answers
2k
views
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 ...
5
votes
1
answer
156
views
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 ...
1
vote
0
answers
37
views
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 ...
0
votes
0
answers
35
views
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 ...
0
votes
1
answer
101
views
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 ...
3
votes
0
answers
92
views
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 ...
1
vote
2
answers
223
views
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 ...
4
votes
2
answers
585
views
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
votes
1
answer
310
views
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 ...
11
votes
1
answer
227
views
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 ...
4
votes
0
answers
58
views
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 ...
1
vote
0
answers
75
views
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 ...
1
vote
1
answer
126
views
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 $\...
1
vote
1
answer
116
views
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 ...
5
votes
1
answer
331
views
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 ...
0
votes
0
answers
109
views
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 ...
0
votes
1
answer
112
views
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
$$g = \frac{...
1
vote
1
answer
67
views
Number of ergodic transverse measures for geodesic laminations - bounded by the genus?
Consider a geodesic lamination $\Lambda$`of a closed hyperbolic surface $S$ of genus $g$, and take a globally transverse closed curve $I$. The lamination induces a return map $R_{\Lambda}: I \to I$, ...
4
votes
1
answer
145
views
Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces
I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
3
votes
0
answers
91
views
Asymmetric minimal surfaces in $H^3$
Inspired by this question, are there any explicit parameterizations of asymmetric minimal surfaces in $\mathbb{H}^3$? E.g. something like the minimal surface which lies over the ellipse given by
$$y^2 ...
5
votes
1
answer
184
views
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 $g\gamma^n=h^m$ ...
5
votes
5
answers
545
views
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
votes
1
answer
443
views
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 :...
1
vote
1
answer
180
views
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, $G\times G$ ...
0
votes
2
answers
108
views
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 ...
7
votes
2
answers
263
views
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 = \...
3
votes
0
answers
92
views
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 ...
0
votes
0
answers
33
views
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 ...
0
votes
1
answer
127
views
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 ...
2
votes
1
answer
178
views
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:...
6
votes
0
answers
342
views
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 (...
0
votes
0
answers
98
views
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 ...
0
votes
0
answers
38
views
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":
$$
0 \leq \lVert x-y-z+w \rVert^2 = \lVert x-y\rVert^2 + \lVert z-w \...
1
vote
0
answers
73
views
Understanding logarithmic law for geodesics
I was reading this seminal paper
https://projecteuclid.org/journals/acta-mathematica/volume-149/issue-none/Disjoint-spheres-approximation-by-imaginary-quadratic-numbers-and-the-logarithm/10.1007/...
12
votes
0
answers
418
views
Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?
I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
1
vote
0
answers
56
views
Monotonicity of root of hyperbolic function
For $\kappa \geq \alpha>0$ and $y \geq 0$, consider the following equation:
$$\sqrt{1-\frac{\alpha }{\kappa }} \tanh \left(y \sqrt{1-\frac{\alpha }{\kappa }}\right)=\tanh \left(y-\frac{\alpha }{2}\...
1
vote
2
answers
144
views
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 ...
2
votes
0
answers
53
views
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(\...
3
votes
0
answers
113
views
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 ...
2
votes
2
answers
184
views
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
votes
0
answers
64
views
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
answers
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 ...