Questions tagged [hyperbolic-geometry]

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73 views

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 ...
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2answers
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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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Dirichlet domains of cusped hyperbolic surfaces

I'm used to think of cusped hyperbolic surfaces as equipped with ideal (all vertices are ideal) fundamental domain, but this is not in general a Dirichlet domain. Is is true that any such surface has ...
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1answer
192 views

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 ...
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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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2answers
178 views

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&...
2
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1answer
110 views

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. ...
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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

Set $K$ 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.,...
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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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1answer
160 views

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 ...
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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 $M$...
4
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1answer
79 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(\...
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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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1answer
271 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 $...
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1answer
159 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 ...
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2answers
291 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 $\...
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1answer
168 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 ...
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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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2answers
211 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 ...
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1answer
81 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 ...
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2answers
633 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 ...
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183 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_{...
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1answer
335 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 $\alpha_1$ ...
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2answers
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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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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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4answers
847 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-...
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1answer
201 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 ...
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1answer
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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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86 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 \{\...
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122 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 ...
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1answer
132 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 ...
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1answer
44 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 ...
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1answer
259 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
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1answer
123 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 ...
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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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1answer
386 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 ...
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1answer
441 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/...
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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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1answer
95 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 ...
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1answer
97 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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2answers
169 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 ...
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73 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 ...
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0answers
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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 ...
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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 ...

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