Questions tagged [hyperbolic-geometry]
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861
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Ideal Ford domain (for finite index subgroup)
Let $G$ be a lattice Fuchsian group with parabolic elements, seen as a discrete subgroup of matrices
$
g=
\begin{pmatrix}
\alpha & \overline{\beta} \\
\beta & \overline{\alpha}
\end{pmatrix}
$...
1
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What is the relationship between $\mathrm{SO}(2)$ and $\mathrm{PSL}(2,\mathbb{R})$?
$\DeclareMathOperator\SO{SO}\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\R{\mathbb{R}}$The holonomy of a hyperbolic surface $S$ in terms of differential geometry is either $\SO(2)$ or $\mathrm{O}(...
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A criterion for loxodromicity in Gromov-hyperbolic spaces
Recall that an isometry of a Gromov-hyperbolic space $X$ is called loxodromic if it has exactly two fixed points on the Gromov boundary $\partial X$, one being "attracting" and the other &...
3
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Coordinate transformation for the hyperbolic plane to the pseudo sphere
There are three common ways to represent the hyperbolic plane. One usually starts with the hyperboloid $x_1^2+x_2^2-x_3^2=-1$ embedded in a Minkowski space with metric $ds^2 = dx_1^2+dx_2^2-dx_3^2$. ...
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Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?
It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
2
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2
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Can a hyperbolic three-manifold have 𝑛 toric boundary components?
I'm wondering if I can construct a class of hyperbolic three-manifolds with $n$ toric boundaries. My idea is to take a bordered hyperbolic Riemann surface $\Sigma_{g,n}$, of genus $g$ with geodesic ...
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Is Tarskian hyperbolic geometry consistent, complete & decidable?
Tarski developed an axiomatic description of Euclidean geometry in first order logic. Its primitive notions are points and its primitive relations are betweeness and congruence of points. The Parallel ...
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References on Hyperbolic Geometry and Teichmuller Theory
I am asking a soft question here.
I am learning hyperbolic geometry on my own. Recently, I have completed the book "Fuchsian Groups" by Svetlana Katok. Also, I have background in Lee's three ...
3
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0
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880
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Definition of quasi-geodesics
I was going through the books Géométrie et théorie des groupes by Michel Coornaert, Thomas Delzant, Athanase Papadopoulos and Metric Spaces of Non-Positive Curvature by Martin R. Bridson, André ...
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References on Riemann surfaces
I have asked the question in MSE, but did not get an answer.
I am asking a soft question here. I am interested in learning about Hyperbolic Geometry. I have read the book named "Fuchsian Groups&...
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Dirichlet region of a free group
Let $G$ be a non-uniform lattice Fuchsian group and let $P$ be a Dirichlet region for $G$. In particular $G$ has parabolic elements, $P$ is not compact and has finite area. We are in the unit disc. Is ...
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Geometrical meaning of a question from Marden
Let $T \in SL(2,\mathbb{C})$ be a normalised Möbius transformation. Then, $$|T(z) - T(w)| =|z-w||T'(z)^{\frac{1}{2}}||T'(w)^{\frac{1}{2}}|$$.
The above is an exercise from Outer Circles by Marden (ex. ...
4
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Random walks on the Poincaré disk
Let $G$ be the group of isometries of the Poincaré disk. Let $\mu$ be a probability measure on $G$, and consider $g_1,..,g_n$ i.i.d. random variables on $G$ distributed according to $\mu$. For $z\in \...
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finding automorphisms of binary hermitian forms
Let $K$ be an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Write $\mathcal{H}(\mathcal{O}_K)$ for the set of hermitian 2 x 2 matrices with entries in $\mathcal{O}_K$. Now, we ...
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Hyperbolic 3-manifolds inside algebraic varieties
I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g.,...
3
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Relating different parametrizations of moduli space of Riemann surfaces
I would like to understand, as explicitly as possible, how different coordinates on the moduli space of Riemann surfaces are related:
On the one hand, there is a parametrization coming from hyperbolic ...
2
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1
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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}...
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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
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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 ...
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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 $M$...
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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(\...
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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 ...
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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 $...
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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 ...
10
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2
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775
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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 $\...
4
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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 ...
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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 ...
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2
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890
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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
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1
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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
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2
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1k
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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 ...
3
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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_{...
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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 $\alpha_1$ ...
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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 ...
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0
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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 ...
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4
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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-...
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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 ...
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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 $\Sigma_g$...
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224
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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 ...
4
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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 ...
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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 ...
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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
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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 ...
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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\...
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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 ...
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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/...
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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 ?
...
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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 ...
0
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129
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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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2
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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 ...
4
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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 ...