# Questions tagged [hyperbolic-geometry]

The hyperbolic-geometry tag has no usage guidance.

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### Mostow rigidity for complex hyperbolic manifolds

A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
Theorem (...

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### Calculating a side in hyperbolic polygon [closed]

if I have a hyperbolic quadrilateral ABCD such that $\angle A=\angle B=\frac{2\pi}{3}$, $AD=BC$ and we know the hyperbolic lengths of sides $AB$ and $CD$, how can I calculate the hyperbolic length of $...

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### fundamental domains in H^2 containing large balls

I would like to construct a genus $g$ surface regularly tiled by triangles (for example by 238 triangles). Edmunds-Ewing-Kulkarni prove that the only obstruction to doing this is Euler characteristic ...

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### Decorated Teichmuller space of a punctured disk and moduli space of the annuls

The decorated Teichmuller space of a disk with n punctures on the boundary and one in the interior is the the space of hyperbolic metrics on such a surface with an extra marking of an horocycle at ...

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### Congruence subgroups in arithmetic lattices of $\mathrm{SO}(n,1)$

I am currently reading the paper Deformation Spaces Associated to Hyperbolic Manifolds by Johnson and Millson, Section 7, and the highlighted bit below has been giving me difficulty:
Specifically, ...

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### Hyperbolic $3$-manifold groups that embed in compact Lie groups

Is there a closed hyperbolic $3$-manifold whose fundamental group is isomorphic to a subgroup of some compact Lie group?
It is known that every surface group can be embedded into any semisimple ...

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### Iteration of a parabolic isometry

Let $a$ be a point of $S^{n-1}$ fixed by a parabolic transformation(an isometry which fixes exactly one point in the boundary and no point in the interior) $\phi$ of $B^{n}$ (conformal ball model). ...

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### Lipschitz property of holonomies fails when stable leaves $W^s(x)$ inside the leaves $W^{ss}(x)$

Let $M$ be compact manifold. suppose $f:M\rightarrow M$ is $C^{2}$.
There is a continuous splitting of the tangent bundle $TM=E^{ss}+E^{s}+E^{u}$ invariant under the derivative $Df$ of the ...

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### Classification of Möbius transformations over the Poincaré disk [closed]

I'm studying the space of relations with given size in the group of isometries of the Poincaré disk. The unitary disk in the complex plane with a suitable metric is the Poincaré disk, and on this ...

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### How was the pair of pants introduced [closed]

There are many results mentioned pairs of pants, and it seems to be a classical model. Why are the pairs of pants so useful?
For example, does it have any application if we estimate the perimeter or ...

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### Gauss Bonnet theorem calculation for pseudosphere

In an attempt to verify Gauss-Bonnet theorem for a Beltrami pseudosphere I calculated a simple case of the Riemann sphere. Am taking curved radius $a$ for geodesic polar co-ordinate from the smooth ...

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### Hyperbolic manifolds containing totally geodesic hypersurfaces which themselves contain totally geodesic hypersurfaces

I will preface this by saying that while I am familiar with the general theory of (semi)-Riemannian manifolds, I am a complete novice when it comes to the specifics of hyperbolic manifolds (I am using ...

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### Why are the medians of a triangle concurrent? In absolute geometry

This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...

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### Which knots are singularities of a hyperbolic cone-manifold structures on $S^3$?

Which knots $K\subseteq S^3$ are such that there is a hyperbolic cone-manifold structure on $S^3$ that has (exactly) $K$ as the singular locus?
What if in Question 1 we restrict the cone angles to be $...

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### Simple Closed Hyperbolic Geodesics on Punctured Spheres

Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...

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### Geodesic current supported on a pencil?

Consider a geodesic current $\mu$ on a closed surface $\Sigma$, as defined by Bonahon ("The Geometry of Teichmüller space via geodesic currents"). These are $\pi_1(\Sigma)$-invariant measures on the ...

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### Intersecting geodesics on a surface from non-intersecting geodesics

Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?

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### delta hyperbolicity of a product of poincare balls? [closed]

I know how to compute the $\delta$-hyperbolicity of a $2D$ poincaré disk of radius 1, but I was wondering how to generalize such computations to:
Higher dimensions
Poincaré disk with radius $r$ and ...

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### Cusps in hyperbolic manifolds and fundamental group

I am reading the book "The Arithmetic of Hyperbolic Three Manifolds" by Maclachlan and Reid and I am having some problems in understanding something about cusps.
The definition they give of a cusp is ...

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### Is the conformal compactification of a convex-cocompact hyperbolic manifold $M$ conformally diffeomorphic to the convex core of $M$?

I am not a hyperbolic geometer, so I apologize if I get anything wrong here, and please correct me.
The conformal compactification $\overline {\mathbb{H}^n}$ of hyperbolic $n$-space $\mathbb{H}^n$ ...

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### First eigenvalue for domains in hyperbolic space

I am interested in examples of bounded open subsets of the hyperbolic space, for which the first eigenvalue of the Dirichlet Laplace operator (acting on functions) is known. In Euclidean space several ...

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### 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 ...

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### Immersions of the hyperbolic plane

Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was ...

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### Question on a proof of density of periodic orbits

In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...

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### Characteristics of Poincare's ball model (of hyperbolic spaces)

Are there any references (papers, books, etc.) that have ready-calculated equations for geometric quantities of the hyperbolic space in terms of the coordinates of Poincare's ball models?
By ...

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### Lattices of PU(n,1) with large abelianization

I am interested in properly discontinuous cocompact subgroups of the group $PU(n,1)$ of automorphisms of the complex hyperbolic space $H^n_{\mathbb{C}}$, says for $n=2,3$. Is there such a lattice $G$ ...

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### Expected distance in hyperbolic space

In a hyperbolic space, $r$ and $\theta$ can represent a point in a polar coordinate system. If we suppose $\theta_1\sim \operatorname{Uniform}(t_1,t_2)$, $\theta_2\sim \operatorname{Uniform}(t_3,t_4)$,...

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### Is the development map in Hyperbolic geometry related to development in Cartan geometry?

I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ ...

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### Cluster algebra and Fenchel Nielsen coordinates

Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the ...

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### How many simple closed geodesics in a given primitive homology class?

It is well-known that an essential closed curve on a hyperbolic surface (possibly with boundary) is homotopic to a unique closed geodesic. Moreover, if the curve under consideration is simple, then so ...

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### Reference request: geometric finiteness of Fuchsian groups

My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong):
Proposition. The convex core of a complete hyperbolic surface ...

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### Four circles on the sphere

Consider 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 enumerate/...

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### $S^3 \setminus S^1$ doesn't have hyperbolic structure

I need to prove that $M = S^3 \setminus S^1$ doesn't admit any metric of constantly negative sectional curvature s.t. $M$ is complete respect for this metric. I know that it is consequence of famous ...

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### What are the $2 \times 2$ matrix generators of $\text{SL}_2\big(\mathbb{Z}[i]\big)(2+i)$?

I have been trying to learn about congruence groups. Here is an example:
\begin{eqnarray*} \Gamma\big(1+2i\big) &=& \text{SL}_2\big(\mathbb{Z}[i]\big)(1+2i) \\ \\
&=& \left\{ \left(...

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### Equality on $\partial \mathbb{H}$ of lifts for isotopy to a conformal map

Let $\mathbb{H} \subset \mathbb{C}$ be the upper half plane. First recall the following statement: if $f^* \colon \mathbb{H} \rightarrow \mathbb{H}$ is quasi-conformal (qc), then there exists an ...

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### Conformal boundary and cusp of figure-8 complement

As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...

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### Intersection of $\pi_1$-injective surfaces

Let $M$ be a closed non-Haken hyperbolic $3$-manifold. Are there two, nonhomotopic, $\pi_1$-injective closed surfaces in $M$, and which do not intersect?

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### 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.
...

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### Harmonic analysis on constant curvature hyperbolic spaces of arbitrary dimension

I am currently looking for a formulation of a Fourier transform on manifolds of constant negative curvature. Specifically, I am looking for generalizations of the two dimensional results on the ...

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### Lifts of geodesics on surfaces onto the universal cover [closed]

self-intersecting geodesic on hyperbolic surface of genus 2
Given a self intersecting geodesic on a hyperbolic surface of genus 2 as in the picture, how can I understand precisely what the lift to ...

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### Immersed incompressible surfaces in surface bundles

Let $M$ be a closed, oriented, hyperbolic $3$-manifold which is a surface bundle over $\mathbb{S}^1$.
Is there some $\pi_1$-injective closed surface (perhaps not embedded) $S \subset M$ which is not ...

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### Hyperbolic structures on infinite type surfaces

Let S be a surface whose fundamental group is NOT finitely generated. Does there exist a complete hyperbolic metric on S for which the area is finite? I suspect the answer in general is NO, but I do ...

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### Are two triangles with equal corresponding medians, congruent?

Is the hyperbolic or spherical analogy of the following Euclidean fact, true?
Two triangles with equal corresponding medians are congruent.
More precisely: Assume that $\Delta ABC$ and $ ...

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### Parabolic subgroups of relatively hyperbolic and CAT(0) groups

Let $G$ be a finitely generated group. We say that $G$ is CAT(0) if it acts properly and co-compactly by isometries on a CAT(0) space.
We say it is hyperbolic relative to a collection $\Omega$ of ...

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### Explanation of an unexplored note of Gauss on hyperbolic volume

My question refers to Gauss's note "Cubirung Der Tetraeder", which can found at p. 228 in the section on the foundations of geometry in volume 8 of his collected works. In this note, Gauss wrote down ...

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### Real section of moduli space of Riemann surfaces

In (https://www.sciencedirect.com/science/article/pii/002240499390049Y) it is mentioned the real section of the moduli space of Riemann surfaces of genus 0. It can be intuitively defined as a subset ...

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### Annuli and pinched annuli vs circles and horocycles

Any annulus is biholomorphich to the Poincare' disk $D$ from wich a circle centered at the origin has been removed.
If we want to parametrise annuli with punctures at one boundary, give the punctures ...

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### Parabolic elements of the Poincare' disk automorphism group as limit of elliptic ones

The automorphism group of the Poincare' disk has elements called elliptic, which have a single fixed point in the interior of the disk, and can be represented as a rotation around this fixed point.
...

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### Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter

First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...

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### Build a Fuchsian group starting from punctures on a disk

Consider the moduli space of hyperbolic metrics on the disk with $n>3$ marked points on its boundary, $\mathcal{M}_{D,n}$.
$\mathcal{M}_{D,n}$ can be parametrised in terms of cross ratios of the ...