Questions tagged [hyperbolic-geometry]
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648
questions
2
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1answer
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Classification of isometries of hyperbolic 3-space
Denote the upper half space by $\mathcal{H}_{3}=\Bbb{C}\times (0,\infty)$. A point $P \in \mathcal{H}_{3}$ is given as, $P=(z, t)=(x, y, t)=z+t j$ where $z=x+i y$ and $j=(0,0,1) .$ The group $P S L_{2}...
24
votes
4answers
817 views
Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...
2
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0answers
25 views
Maximum entropy distribution in the hyperbolic plane with given “mean” and “variance”
On the Euclidean sphere, by defining suitable analogs of the mean and variance in terms of the first circular moment, it can be shown that the von Mises distribution is the maximum entropy ...
7
votes
1answer
135 views
Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?
For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
9
votes
1answer
238 views
Cutting up the Bring surface into six pairs of pants
The Bring sextic, with 120 automorphisms, is the numerically most symmetric compact Riemann surface of genus 4. To cut it up into six pairs of pants, we need to cut along nine disjoint geodesic loops....
4
votes
1answer
270 views
When does a subgroup of $\operatorname{GL}(n, \mathbb Q)$ have a bounded fundamental domain on $\mathbb R^n$?
$\DeclareMathOperator\GL{GL}$Let $G \subset M_{n\times n~}(\mathbb Z)$ be a finitely generated subgroup of $\GL(n,\mathbb Q)$ (i.e. $g\in G$ is an invertible matrix with entries in $\mathbb Z$). Then $...
4
votes
0answers
31 views
Covering hyperbolic manifolds by round balls
This question is a natural follow up to the question asked here. I think it should not be too hard to answer it (negatively) in dimension 3, but higher dimensions will be probably challenging.
Let $M$...
10
votes
2answers
544 views
Can you cover a genus a billion hyperbolic surface with 15 balls?
Here's a question I was wondering about this week. Not sure how interesting it is, but I thought it was kind of curious.
Question: Given $k$, is there a number $N=N(k)$ such that if a closed ...
4
votes
1answer
73 views
Is the space of Euclidean polyhedra with a fixed $1$-skeleton connected?
Let $\mathcal{A}_\Gamma$ be the space of convex (non-degenerate) Euclidean polyhedra with $1$-skeleton a certain polyhedral graph $\Gamma$. This space can be seen as a subset of $\mathcal{Gr}_2(\...
6
votes
2answers
163 views
Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider:
The ...
1
vote
1answer
146 views
Fixed point free involutions on Riemann surfaces
It is well known that a Riemann surface can have a fixed point
free holomorphic involution only if it has odd genus. If it has one, is it unique?
More generally, is any fixed point free automorphism ...
4
votes
1answer
165 views
Mapping the hyperbolic plane onto the interior of a disk
In Chapter II of his book Non-Euclidean Geometry (1961; first published in Polish, 1956), Stefan Kulczycki defines a mapping of the hyperbolic plane onto the interior of a disk. Its construction ...
10
votes
2answers
280 views
Optimal $\delta$ for Gromov's $\delta$-hyperbolicity of the hyperbolic plane
What is the minimal $\delta$ such that the hyperbolic plane is $\delta$-hyperbolic, in the sense of the four point definition of Gromov?
Four point definition of Gromov: A metric space $(X, d)$ is $\...
5
votes
2answers
2k views
Triangle area on surfaces of constant curvature
I am looking for an elementary derivation of the formula for the area of a geodesic triangle lying in a surface of constant curvature $\kappa$, depending on the angles and side length.
Of course, the ...
Alexandre Eremenko
67.6k6 gold badges194 silver badges324 bronze badges
1
vote
2answers
601 views
Explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface
I am finding the explicit description of genus $2$ surface as the upper half plane modulo group of Deck Transformation. I did't find it anywhere. Also, I found a similar question here. But, there is ...
21
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6answers
2k views
Tetrahedra with prescribed face angles
I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/...
1
vote
0answers
36 views
How to tile a plane such that moving from one tile to the next in any of the 8 cardinal directions is the same length?
When tiling the euclidean plane with squares (like most board games), moving diagonally to another tile is longer than moving vertically or horizontally. Is there a tiling such that moving in any of ...
1
vote
2answers
192 views
Cross ratio in hyperbolic geometry
In the rough sketch four concurrent lines are drawn in the Poincaré disk model and in the Euclidean model.
If same angles $ (\alpha,\beta,\gamma,\delta) $ are enclosed at respective points of ...
2
votes
1answer
75 views
Hyperbolic length of curve that does not enter a collar
Let $\Sigma$ be a compact surface of genus at least $1$ with one boundary component, equipped with a hyperbolic metric so that the boundary is geodesic. There is a version of the collar lemma that ...
3
votes
0answers
181 views
Hyperbolic metrics and the general Ahlfors-Bers theorem
Let $M$ be an oriented smooth compact 3-manifold with non-empty boundary and hyperbolizable interior such that all boundary components have genus greater than $1$. Denote $N:={\rm int}(M)$ and
$$HM_{...
3
votes
1answer
367 views
What is the representation of the generators of the triangle group for the uniform (4 4 4) tiling of hyperbolic disk as Mobius transformations?
I wonder how can one describe the generators of the triangle group for the tesselation of Poincare unit disk by triangles with angles $\pi/4, \pi/4 , \pi/4 $ in terms of the action of the modular ...
8
votes
2answers
206 views
Gromov hyperbolicity constant vs. Gromov-Hausdorff distance to a tree
Let $X$ be a compact, geodesic metric space which is Gromov hyperbolic with a constant $\delta>0$. To fix scaling, let us also assume that $X$ has diameter $1$.
To fix a definition of Gromov ...
5
votes
1answer
326 views
Statements related to Thurston's work on the surface
If we have simple closed curves $\alpha$ and $\beta$ on a surface $\Sigma_g$, the intersection number $i(\alpha ,\beta)$ is defined to be the minimal cardinality of $\alpha_1\cap\beta_1$ as $\alpha_1$ ...
4
votes
1answer
127 views
Absolute and relative tilings of the hyperbolic plane
In Conway's Symmetries of Things on p. 265 I found these two tilings of the hyperbolic plane with the same vertex configuration $(3.5.3.5.3)$ (resp. vertex figure, as Conway calls it).
The ...
0
votes
0answers
73 views
Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$
I need an algorithm to efficiently find at least one lattice point on a hyperbola of equation $axy+bx+cy+d=0$.
Lattice point means integer coordinates and equation with integer means diophantine ...
3
votes
1answer
200 views
What is the name of this geometric structure, where we identify each sphere of vision with the sphere at infinity?
If you consider hyperbolic $n$-space $H^n$, modeled by the open unit ball $B^n \subset \mathbb{R}^n$, then given any two distinct points $x_1$, $x_2$ in $H^n$, there is a natural way of identifying ...
4
votes
1answer
78 views
Explicit check of the invariance of the Weyl-Petersson form
Using Fenchel-Nielsen coordinates, the Weyl-Petersson metric can be written as
$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$
where $i$ is an index labelling the curves of a pants decomposition ...
9
votes
1answer
119 views
When do the lengths of simple closed curves determine a hyperbolic surface?
Consider hyperbolic metrics on $\Sigma_g$ a closed orientable surface of genus $g$. Let $[\gamma_1] , \cdots, [\gamma_n]$ be a finite collection of isotopy classes of simple closed curves on $\Sigma_g$...
Alexandre Eremenko
67.6k6 gold badges194 silver badges324 bronze badges
0
votes
0answers
82 views
2 dimensional Hausdorff measure and area measure on the hyperbolic plane
$\newcommand{\diam}{\operatorname{diam}}$From the Encyclopedia of Mathematics, the Hausdorff measure on a generic metric space (X,d) can be defined using
$H^\alpha_\delta (E):=\omega_\alpha \inf \{\...
1
vote
0answers
117 views
Exponential map and optimization
Apologies in advance for being somewhat vague. I'm trying to get pointers to establish a connection between a common trick used in practice in optimization, and the exponential map in differential ...
22
votes
1answer
438 views
Alternate proofs that hyperbolic plane can’t be isometrically immersed in $\mathbb{R}^3$
A famous theorem of Hilbert says that there is no smooth immersion of the hyperbolic plane in 3-dimensional Euclidean space. The expositions of this that I know of (in eg do Carmo’s book on curves/...
0
votes
1answer
39 views
What is sequential boundary of a $\delta$-hyperbolic space and how is the Gromov product extended to the boundary?
I have been reading up on $\delta$-hyperbolic spaces. But I am not getting a clear idea of sequential boundary of $\delta$-hyperbolic spaces and how the Gromov product is extended to it. Could ...
7
votes
1answer
239 views
Complete geodesics on hyperbolic a pair of pants
I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:
I am trying to understand the article by Maryam ...
3
votes
1answer
121 views
Cusps of hyperbolic surfaces under finite covers
The following statement seems true, but I don't know a proof or a reference for it (and I would like one).
Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We ...
0
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1answer
95 views
Bounded subsets of $\delta$-hyperbolic metric spaces
I was reading this book by Coorneart, Delzant, and Papadopoulos. I am stuck with this proposition in Chapter 1 (Proposition 1.5)
If $Y$ is a bounded $\delta$-hyperbolic subset of $X$, then $X$ is $\...
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vote
0answers
47 views
Weil-Petersson metric with respect to covering
Let $S$ be a closed oriented surface of genus $g\geq 2$. Consider the Teichmuller space $T(S)$. Let $d_t$ be the Teichmuller metric and $d_{WP}$ be the Weil-Petersson metric on $T(S)$. Let $P:S_1\...
3
votes
2answers
431 views
How to derive from Gauss's result on the volume of hyperbolic orthoscheme tetrahedron the formulas of Lobachevsky and Bolyai?
My question is a direct continuation of my already posted question https://hsm.stackexchange.com/questions/6772/did-gausss-expression-for-the-differential-of-the-hyperbolic-volume-of-the-tetr (asked ...
3
votes
0answers
76 views
Presentations of $\mathbf{PGL}_3(\mathbb{F}_q)$ by three involutions
If I am not mistaken, the group $\mathbf{PGL}_3(\mathbb{F}_q)$, with $\mathbb{F}_q$ the finite field with $q$ elements, can be generated by three involutions.
Where could I find such representations ?
...
1
vote
1answer
93 views
Presentations of $\mathbf{𝐏𝐆𝐋}_3(\mathbb{F}_2)$ by three involutions, 2
I am searching for a presentation of the group $\mathbf{PGL}_3(\mathbb{F}_2)$ for which the generators are involutions $a, b, c$, and such that the following relations are present [among extra ...
4
votes
0answers
72 views
What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?
Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
3
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0answers
74 views
An explicit (maybe algebraic) isometric embedding of the double torus with constant curvature K = -1
The following question is related to this previous question, Canonical immersion of the double torus:
Is there any known explicit (maybe algebraic) isometric embedding of a genus 2 surface endowed ...
20
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1answer
656 views
Canonical immersion of the double torus
It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
6
votes
1answer
102 views
Number of Fuchsian groups with same trace field
Let $\Gamma,\Sigma\subset \mathrm{SL}_2({\mathbb R})$ be cocompact arithmetic subgroups. They are called commensurable in the wider sense, if there exists
$g\in \mathrm{SL}_2({\mathbb R})$, such that ...
1
vote
0answers
69 views
Continuity of conformal grafting, wrt the (infinite type) surface
Say I have two closed hyperbolic surfaces $X,Y$ and a smooth, $(1+\epsilon)$-bilipschitz map $f : X \to Y$ for some small $\epsilon$. Pick a simple closed curve $c \subset X$, and let $X',Y'$ be the ...
4
votes
0answers
106 views
Hyperbolicity of twist knots
In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the ...
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votes
0answers
29 views
Invariants in spherical and pseudospherical triangle trigonometry
We know the spherical surface invariant in $\mathbb R^3$
$$ \dfrac{\sin a }{\sin A } =\dfrac {\sin b }{\sin B } =\dfrac{\sin c }{\sin C } =\dfrac{\sqrt{1-\cos^2 A -\cos^2 B-\cos^2 C +2\cos A \cos B \...
7
votes
0answers
106 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
0answers
97 views
What does hyperbolicity of curves buy us in the arithmetic context?
This is going to be a fairly vague question but hopefully it will have concrete answers:
There is this recurrent phenomenon in the arithmetic of curves (possibly stacky,affine) where there is a "...
2
votes
0answers
79 views
Bounds on convergence of two orbits in the limit set of a Schottky group
Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...
3
votes
1answer
119 views
Maximally symmetric hyperbolic 3-manifolds with finite volume
In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...
