Questions tagged [hyperbolic-geometry]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
1 answer
150 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 ...
9 votes
2 answers
3k views

What is / are the softwares to use to draw surfaces of the form of a two or three-holed torus , or torus, or torus with cusps attached to it?

I am trying to draw surfaces with complete hyperbolic structures and surfaces which are topologically tori. The hyperbolic surfaces I need to draw are torus with one or two holes on it, or torus with ...
5 votes
1 answer
185 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 $g\gamma^n=h^m$ ...
3 votes
0 answers
93 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 ...
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, $G\times G$ ...
40 votes
1 answer
1k views

Four circles on the sphere

Consider generic configurations consisting of 4 distinct circles on the sphere. Two configurations are equivalent if they can be mapped onto each other by a homeomorphism of the sphere. How to ...
0 votes
1 answer
128 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 ...
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 ...
2 votes
1 answer
200 views

Lamination as limit of arcs

I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
2 votes
2 answers
299 views

Why simple closed curves are dense in $\mathcal{PML}_0(S)$?

I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...
4 votes
2 answers
195 views

Measured geodesic laminations have either discrete or Cantor set local cross-sections

I'm reading through Kerckhoff's paper "The Nielsen Realization Problem": https://www.jstor.org/stable/2007076. In section 1, after he defines measured geodesic laminations, he makes the ...
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 ...
2 votes
1 answer
179 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:...
0 votes
0 answers
103 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
75 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
146 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 ...
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 ...
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
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(\...
6 votes
1 answer
1k views

How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?

My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
18 votes
3 answers
1k views

How to see isometries of figure 8 knot complement

The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
2 votes
2 answers
185 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 ...
13 votes
0 answers
195 views

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 ...
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 ...
9 votes
0 answers
235 views

Systole of Riemann surfaces of genus $g$

In Buser and Sarnak's "On the period matrix of a Riemann surface of large genus", we get $$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
4 votes
1 answer
188 views

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 views

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) ...
1 vote
1 answer
109 views

Does the rational normal curve embedding extend as a mapping from the "bulk" to some bigger ambient space?

The complex projective line $\mathbb{P}^1(\mathbb{C})$ can be identified with the sphere at infinity of hyperbolic $3$-space, modeled say by the Poincare open $3$-ball in $\mathbb{R}^3$ (the sphere at ...
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 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 $\pi(K) = \pi_1(S^3 \setminus K)$...
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 ...
9 votes
1 answer
452 views

What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary (...
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 ...
13 votes
2 answers
960 views

Measure on the Boundary of a Hyperbolic Group

Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance ...
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 ...
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 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'...
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/...
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 ...
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 ...

1
2
3 4 5
18