Questions tagged [hyperbolic-geometry]

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Cluster algebras of type A and X

I will base my question on Fock and Goncharov's paper Dual Teichmüller and lamination spaces. Let $S$ be a surface with boundaries, marked points on such boundaries, punctures and boundaries without ...
giulio bullsaver's user avatar
4 votes
0 answers
153 views

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 \...
Chevallier's user avatar
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0 answers
87 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 ...
Calvin McPhail-Snyder's user avatar
4 votes
0 answers
119 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 "...
Asvin's user avatar
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126 views

Area lower bound given a mean curvature upper bound?

If $\Sigma$ is a smooth embedded closed hypersurface in $\mathbb R^n$ with (normalized) mean curvature $H\le 1$ (the mean curvature of the unit sphere), then its ($(n-1)$-dimensional) area is at least ...
Nobody's user avatar
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283 views

Bers' simultaneous uniformization

I have been trying to understand Bers' famous paper "Simultaneous Uniformization". Regarding this paper I have a few questions. Any kind of help will be appreciated. Let $S$ and $S^{'}$ be two ...
P.S's user avatar
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206 views

Explanation for phenomenon in hyperbolic geometry

By examining numerous examples I have become quite convinced that the following statement is true. Let $x$ and $y$ be distinct points in the hyperbolic plane $\mathbb{H}$. Let $\gamma$ be the ...
burtonpeterj's user avatar
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Quadrics in $\mathbb{H}^3$

Consider a hyperbolic space $\mathbb{H}^3$ in the Beltrami-Klein model: $$\mathbb{H}^3=\{(x,y,z|x^2+y^2+z^2\leq 1\}\subset \mathbb{R}^3.$$ Let $Q$ be a quadric in $\mathbb{R}^3.$ Question: What is a ...
Daniil Rudenko's user avatar
4 votes
0 answers
251 views

Hyperbolic Intercept (Thales) Theorem

Is there an Intercept theorem (from Thales, but don't mistake it with the Thales theorem in a circle) in hyperbolic geometry? Euclidean Intercept Theorem: Let S,A,B,C,D be 5 points, such that SA, SC, ...
tisydi's user avatar
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Given a CAT(0) space can one construct a CAT(-1) space that has the other space as boundary?

Two easy questions: Given a metric space $(Z,d_Z)$, let $X = \text{Con}_h(Z) = Z \times ]0,\infty[$ and for $(z,i), (z',i') \in \text{Con}_h(Z)$ define a metric on $X$ as follows: $d((z,i), (z',i')) =...
Loreno Heer's user avatar
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600 views

Visual boundary vs. ideal boundary of hyperbolic manifolds?

I apologize in advance if this is elementary; I have very little experience with hyperbolic manifolds, but I'm using hyperbolic manifolds for part of a current project. Given a discrete torsion-free ...
ಠ_ಠ's user avatar
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Exponential contraction for the projection on horospheres

A few years ago, Roberto Frigerio asked for a reference for a geometric property of horospheres (Reference for the geometry of horospheres), namely exponential decay of the projection onto a ...
M. Dus's user avatar
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Is there a formula for the A-model partition function in terms of hyperbolic structure?

The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant? I'm not sure if there is a way to normalize these since the usual perscription $Z(S^2) ...
user404153's user avatar
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Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions. By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in $...
Xin Nie's user avatar
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Origin of spectral theory on infinite-area hyperbolic surfaces

The study of spectral theory of finite-area hyperbolic surfaces is intimately related to number theory, in particular by the importance of Maass cusp forms. The counting of resonances is of importance,...
Sabine Müller's user avatar
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299 views

Haken manifolds and characterising sutured manifold hierarchies

In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--...
Don Shanil's user avatar
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reference request: “p-adic” presentation of surfaces

On several occasions I heart about the following result: For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
Roger Weilik's user avatar
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0 answers
93 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 ...
Edoardo Rizzi's user avatar
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 ...
JMK's user avatar
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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 ...
bergfalk's user avatar
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 ...
Justin_other_PhD's user avatar
3 votes
0 answers
67 views

Discreteness of volumes of boundary-parabolic representations

Suppose $M$ is a cusped hyperbolic $3$-manifold of finite volume. Let $\mathfrak{R}_0(M)$ be the space of boundary-parabolic representations $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb C)$. Is ...
Calvin McPhail-Snyder's user avatar
3 votes
0 answers
335 views

Representations of triangle groups

$\DeclareMathOperator\SU{SU}\DeclareMathOperator\PSL{PSL}$I am self-studying triangle groups and the following question comes up. Let $G$ denotes $(2,3,7)$ triangle group. It is symmetry group of $(2,...
KAK's user avatar
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0 answers
148 views

Variants of Selberg trace formula

I am familiar with a basic case of Selberg's trace formula, in the case of quotients of upper half plane (for example, see Sections 5.1 - 5.3 of Bergeron's book). Section 5.1 describes a general setup ...
user482438's user avatar
3 votes
0 answers
221 views

Immersion of a part of the hyperbolic plane in $\mathbb{R}^3$

I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert's theorem ensures that there is no isometric immersion ...
Zaragosa's user avatar
  • 123
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0 answers
892 views

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 ...
Alex George's user avatar
3 votes
0 answers
100 views

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 ...
giulio bullsaver's user avatar
3 votes
0 answers
223 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_{...
Roman's user avatar
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0 answers
80 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 ? ...
THC's user avatar
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3 votes
0 answers
101 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 ...
Mauro Patrão's user avatar
3 votes
0 answers
149 views

Riemannian metric over moduli space of Riemann spheres with n punctures

In the paper `Tessellations of moduli spaces and the mosaic operad' by Devadoss (https://arxiv.org/pdf/math/9807010.pdf), on page 5-6, the author identifies hyperbolic planar tree space (or the ...
thienle's user avatar
  • 31
3 votes
0 answers
98 views

Existence of eigenvalues and eigenvalues of infinite multiplicity in geometrically finite manifolds with infinite volume

In the paper, The geometry and spectra of hyperbolic manifolds https://link.springer.com/article/10.1007/BF02830802 by PETER D HISLOP, the author sketched a proof for the following theorem: Let $\...
Hans's user avatar
  • 113
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0 answers
223 views

What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$?

What is the connection between $\mathrm{AdS}_2$ and the hyperbolic plane $\mathbb{H}^2$? Some sources seem to imply that they are the same, i.e. having at least the same symmetry group $\mathrm{SL}(2,...
horropie's user avatar
  • 649
3 votes
0 answers
137 views

Aperiodic tile with rational area

Margulis and Mozes constructed aperiodic tiling system on the hyperbolic plane consisting of a single tile(hyperbolic polygon) whose area (or each inner angle) is irrational multiple of $\pi$. Having ...
Arun 's user avatar
  • 725
3 votes
0 answers
108 views

Density of closed orbits on hyperbolic surfaces

It is well-known that the set of closed geodesics on a closed hyperbolic surface is dense. My questions: If this property still holds on finite-area hyperbolic surfaces, infinite-area hyperbolic ...
user avatar
3 votes
0 answers
325 views

The uniqueness of Poincaré metric

The Poincaré metric $ds=\frac{\sqrt{dx^2+dy^2}}{y}$ has the proprety that the action of the group $PSL(2,\mathbb{R})=SL(2,\mathbb{R})/\{\pm I_{2}\}$ on $\mathbb{H}$ preserves the hyperbolic distance. ...
geometer13's user avatar
3 votes
0 answers
70 views

Does the orbital function divided by the volume of a ball decrease?

Let $X$ be a Cartan-Hadamard manifold, meaning a complete, connected, simply connected Riemannian manifold with non-positive sectional curvature and $\Gamma < Isom(X)$ a discrete group of ...
user avatar
3 votes
0 answers
188 views

Ending lamination theorem

Let $M$ a compact manifold with surfaces $S_1,...,S_p$ as boundaries. Let us suppose that $M$ admits a complete hyperbolic structure. Then, from the ending lamination theorem, given either laminations ...
user avatar
3 votes
0 answers
230 views

Pairs of non-isometric subsurfaces of a hyperbolic 3-manifold, with the same genus

When I say manifold below, I mean a complete orientable finite-volume hyperbolic $3$-manifold and when I say subsurface, I mean immersed closed totally geodesic subsurface. Whenever a manifold has a ...
j0equ1nn's user avatar
  • 2,438
3 votes
0 answers
152 views

Question on Neumann-Zagier Symplectic matrix of an ideal triangulation of one cusped 3-manifold

For a knot complement $M\backslash K$ on a general closed 3-manifold $M$, the gluing equations are given by ($k$ is the number of ideal tetrahedra in an ideal triangulation of $M\backslash K$) $c_I:=\...
강동민's user avatar
  • 115
3 votes
0 answers
321 views

Curvature $\geq-1$ but not $\geq1$

(Edited again) In the following, for brevity, I will say that $$X\ \ \mathrm{has}\ \ \kappa_{\mathrm{max}}=k$$ if $X$ is a compact ($n$-dimensional with $n\geq2$, with empty boundary) Aleksandrov ...
stefaNo's user avatar
  • 31
3 votes
0 answers
48 views

Geodesics in norm balls

Recently, some problems that I work on require that I understand a bit of hyperbolic complex geometry. Assume that $B \subset \mathbb{C}^n$ is the unit ball of some norm $\|\cdot\|$ (not induced by an ...
shamovic's user avatar
  • 431
3 votes
0 answers
405 views

Geometric intersection number for product of elements of the fundamental group

Let $F$ be a hyperbolic surface and $p\in F$ be a point. Consider $\pi_1(F,p)$, the fundamental group of $F$ with base point $p$. Let $x,y\in \pi_1(F,p)$ and $z$ be a simple closed curve in $F$ such ...
Cusp's user avatar
  • 1,703
3 votes
0 answers
196 views

Uniform continuity of length function on geodesic currents

I'm starting to study geodesic currents and I have a question concerning uniform continuity. Let's take $S$ a closed surface of genus $g$ and $GC(S)$ the space of geodesic currents on $S$ (as it is ...
user3419's user avatar
3 votes
0 answers
82 views

Length and laplacian spectrum for quasi-fuchsian manifold

It is well known that, in the case of finite area hyperbolic surfaces, the length sprectrum (the collection of length of all closed geodesics) and the spectrum of the laplacian (acting on functions) ...
user avatar
3 votes
0 answers
198 views

$\mathbb{CP}^1$-structures and hyperbolic Gauss maps

Let $\Sigma$ be a closed surface of genus at least $2$. Put a quasi-Fuchsian $\mathbb{CP}^1$-structure (i.e. complex projective structure) on $\Sigma$. Thus the universal cover $\tilde{\Sigma}$ is ...
Xin Nie's user avatar
  • 1,764
3 votes
0 answers
339 views

Discussion of specific arithmetic triangle groups?

Arithmetic triangle groups were classified in Takeuchi, Arithmetic triangle groups, J. Math. Soc. Japan Volume 29, Number 1 (1977), 91-106. The (2,3,7) case was discussed in detail in a number of ...
Mikhail Katz's user avatar
  • 15.1k
3 votes
0 answers
224 views

The distance between two farthest points on the Bolza surface?

The Bolza surface $M$ is the closed hyperbolic surface of genus $2$ that can be obtained by identifying the opposite sides of the regular octagon in $\mathbb{H}^2$. What two points on $M$ are ...
Mikhail's user avatar
  • 31
3 votes
0 answers
415 views

Boundary defining functions for hyperbolic surfaces

Let $M$ be a geometrically finite hyperbolic surface with one cuspidal end and one funnel end so that it can be divided into $C \cup K \cup F$ where $C$ is the cusp, $F$ the funnel and $K$ the ...
Robin Neumann's user avatar
3 votes
0 answers
172 views

Collapsing the medial axis of a polytope

Let X be a convex polyhedron in hyperbolic 3-space. Let M be the medial axis of X. Question: Is M collapsible? It is easy to see that M is contractable. In the case of Euclidian 3-space, instead ...
Shinpei 's user avatar