# Questions tagged [hyperbolic-geometry]

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154
questions with no upvoted or accepted answers

**23**

votes

**1**answer

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### Examples of Einstein four-manifolds of negative sectional curvature

Are there any nontrivial compact Einstein four-manifolds of negative or nonpositive sectional curvature? by nontrivial we mean not quotients of $\mathbb{H}^4$, $\mathbb{C}H^2$, $\mathbb{H}^2\times\...

**17**

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**0**answers

517 views

### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...

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760 views

### How does duality of symmetric spaces explain the hyperbolic cosine theorem?

There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...

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337 views

### What is the cup-product structure like on a hyperbolic 5-manifold?

Let $X$ be a hyperbolic 5-manifold. Can there be any class in $H^2(X;\mathbf{Z})$ that is torsion and whose square in $H^4(X;\mathbf{Z})$ is not zero?
For example, are there hyperbolic 5-manifolds ...

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282 views

### More mysterious properties of Gram matrix

This is another question related to the mysterious properties of the Gram matrix in dimension $4$. Here's the previous question.
The following fact could be extracted from 0402087:
For any $a_i\...

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309 views

### Right-angled polytopes

%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not exist ...

**10**

votes

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215 views

### How the hyperbolic metric changes when we add a puncture?

Suppose we have a surface of a finite genus, without boundary with a finite number of punctures. This surface admits a unique hyperbolic metric of curvature $-1.$
If I add a puncture somewhere, the ...

**10**

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317 views

### Hyperkähler structure on the moduli space of tetrahedra?

Consider a moduli space of geodesic tetrahedra in the hyperbolic space $\mathbb{H}^3.$ In the Klein's model the hyperbolic space can be presented as the interior of a unit ball
$$
\mathbb{H}^3=\{(x_1,...

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161 views

### 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 ...

**10**

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118 views

### Compatibility of spherical and hyperbolic geometry for fibred knots

Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...

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269 views

### Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$

Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...

**10**

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471 views

### Is there a general dilogarithm formula for the Cheeger--Chern--Simons class?

I'm looking for a generalization of the calculation of the hyperbolic volume and Chern--Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.
...

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295 views

### Phillips-Sarnak conjecture in higher dimension

The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...

**8**

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289 views

### Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...

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254 views

### Lines in upper half-space

A couple of years ago, I taught an undergraduate class introducing various aspects of classical geometry, learning the (beautiful!) subject as I went. I found one thing that really bothered me: the ...

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169 views

### Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?

"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...

**7**

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109 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 ...

**7**

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119 views

### Purely analytic proof of the Nielsen-Thurston classification theorem

I hope this question is appropriate for the site. I've been looking at the expositions of Bers' proof of the Nielsen-Thurston classification given in Hubbard's Teichmüller Theory
and Applications to ...

**7**

votes

**0**answers

987 views

### Closed geodesics on a closed, negatively curved Riemannian manifold

I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...

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357 views

### Does every hyperbolic 3 manifold with totally geodesic boundary has some finite covering space with more than one boundary component?

I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\
The ...

**6**

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364 views

### Embed the hyperbolic plane into Euclidean spaces

Can the complete simply-connected surface with constant Gauss curvature -1 be embedded smoothly in the 5-dimensional Euclidean space?
Can the complete simply-connected surface with constant Gauss ...

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76 views

### "There exists $e_0(S)$ such that the shortest nonperipheral curve on $(S, x)$ has extremal length at most $e_0$

I was reading the paper by Masur-Minsky (Geometry of the Complex of Curves I: Hyperbolicity) where they show the curve complex $C(S)$ to be $\delta$-hyperbolic. There given a surface $S$ with ...

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251 views

### Rational stable translation length

Let $G$ be a finitely generated group and $S$ a finite generating set and consider the word metric associated to $S$.
If $g\in G$, define its stable translation length as $l(g)=\lim_n \frac{d(e,g^n)}{...

**6**

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174 views

### Stable norm on hyperbolic surfaces

For a hyperbolic surface $S$ and a homology class $h\in H_1(S)$ its stable norm is defined as $\lim_{n\to\infty}\frac{1}{n}l(nh)$, where $l(nh)$ means the minimal length among all closed geodesics ...

**6**

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228 views

### Locally symmetric spaces: spectrum of the Laplacian

Let $M = \Gamma\backslash X$ denote a locally symmetric space of non-compact type and $\Delta$ the Laplacian on $L^2(M)$.
It is known that the spectrum of $\Delta$ decomposes into finitely many
...

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401 views

### Questions on Thurston's metric on Teichmüller space

I'm reading the famous "Minimal stretch maps between hyperbolic surfaces" by William Thurston and I'm trying to understand the key theorem 8.1. I have many unclear points so I hope someone can help me ...

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184 views

### Is there any exotic smooth structure on open hyperbolic manifold?

I edited my post to clarify some confusions as suggested by Igor.
Let $M$ be an open hyperbolic manifold, with or without finite volume, Is there any manifold $N$ which is homeomorphic to $M$ but ...

**6**

votes

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185 views

### A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers

I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that
a) the graph $\Gamma$ is fine,
b) $\Gamma$ is not a tree,
c) not all ...

**6**

votes

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141 views

### Maximum relator and hyperbolicity

It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows:
Suppose ...

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375 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

**6**

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194 views

### Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...

**6**

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367 views

### A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture:
"... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...

**5**

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167 views

### Effect of the inverse exponential map on the curvature of a given curve

Suppose you have a curve $\alpha$ in a manifold $\mathcal{M}$. You are at a point $\alpha(t)$ of that curve. The curvature of $\alpha(t)$ is the same as the curvature of the curve $exp^{-1}_{\alpha(t)}...

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133 views

### Finitely generated nilpotent groups as cusp groups

I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...

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179 views

### Barycentric interpolation in hyperbolic triangles

Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...

**5**

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258 views

### Reference request: 3-dimensional Mobius transforms

I am working on a project that I suspect requires calculations involving Möbius transformations on 3-dimensional space $\mathbb{R}^3$, identified with the quaternions $\mathbb{H}$ with $k$-component ...

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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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57 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 \...

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33 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 $M$...

**4**

votes

**1**answer

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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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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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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99 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 "...

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44 views

### Smoothness of boundary of $r$-neighborhood of convex core

The boundary of an $r$-neighborhood of the convex core of hyperbolic $n$-manifold is smooth, by page 73 of Hyperbolic Manifolds and Kleinian Groups. The authors does not provide a proof for this fact. ...

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110 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 ...

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131 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 ...

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197 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 ...

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61 views

### Converse to Wolpert's Lemma

Recall Wolpert's lemma:
Let X,Y be hyperbolic surfaces and $f:X\to Y$ a $K$-quasiconformal homeomorphism. For any homotopy class of curves $c$ let $\ell(c)$ denote the length of the geodesic in the ...

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177 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, ...

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65 views

### 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')) =...