Questions tagged [hyperbolic-geometry]
The hyperbolic-geometry tag has no usage guidance.
220
questions with no upvoted or accepted answers
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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, ...
15
votes
0
answers
887
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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 ...
13
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0
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195
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Are there free and discrete subgroups of $\mathrm{SL}_2(\mathbb{R}) \times \mathrm{SL}_2(\mathbb{R})$ that are not Schottky on any factor?
$\DeclareMathOperator\SL{SL}$Does there exist a free and discrete subgroup $\Gamma < \SL_2(\mathbb{R}) \times \SL_2(\mathbb{R})$ such that neither $\pi_1(\Gamma)$ nor $\pi_2(\Gamma)$ is free and ...
13
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0
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364
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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 ...
12
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0
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418
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Is there a mistake in Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions"?
I am having troubles with Don Zagier's "Hyperbolic manifolds and special values of Dedekind zeta-functions", available at this link, and I think there might be a mistake. In particular the ...
12
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0
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454
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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\...
11
votes
0
answers
255
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 ...
11
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0
answers
344
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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
0
answers
369
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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
$$
\mathbb{H}^3=\{(x_1,...
10
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0
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127
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Compatibility of spherical and hyperbolic geometry for fibred knots
Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other ...
10
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279
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Embeddings of hyperbolic $n$-manifolds in $R^{n+2}$
Is there any example of a compact manifold $M$ of dimension $n>10000$
such that
$M$ admits an embedding into $\mathbb R^{n+2}$,
$M$ is hyperbolic; i.e., it admits a Riemannian metric with
...
9
votes
0
answers
235
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Systole of Riemann surfaces of genus $g$
In Buser and Sarnak's "On the period matrix of a Riemann surface
of large genus", we get
$$\frac4{3}\le\limsup_{g\rightarrow\infty}\frac{\max\{\operatorname{sys}(S)|S\in\mathcal{M}_g\}}{\log ...
9
votes
0
answers
326
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, ...
9
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0
answers
346
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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
votes
0
answers
93
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Is the hypotenuse operation associative in every Tarski plane?
By a Tarski space I understand a mathematical structure $(X,\mathsf B,\equiv)$ consisting of set $X$, a betweenness relation $\mathsf B\subseteq X^3$ and a congruence relation ${\equiv}\subseteq X^2\...
8
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0
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242
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What does the Chern-Simons invariant of a hyperbolic $3$-manifold mean?
Let $M$ be a closed $3$-manifold and $\rho : \pi_1(M) \to \operatorname{SL}_2(\mathbb C)$ a representation.
(Feel free to replace $\rho$ with a flat $\mathfrak{sl}_2$ connection with holonomy $\rho$.)
...
8
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0
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374
views
The figure eight knot complement in $S^3$
Recently I have been going through the book Hyperbolic Knot Theory by Jessica Purcell. Exercise 5.4 (on page 101) gives us a presentation of the fundamental group of $S^3 - K$ where $K$ is the figure-...
8
votes
0
answers
920
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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 ...
8
votes
0
answers
306
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 ...
8
votes
0
answers
189
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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
votes
0
answers
198
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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
331
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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
...
7
votes
0
answers
1k
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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 ...
7
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0
answers
373
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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
votes
0
answers
342
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Why can't a Lie group act transitively on a finite volume hyperbolic manifold?
In the comments on the MathSE question "Is Seifert-Weber space homogeneous for a Lie group?",
it is claimed that if $ M $ is a manifold which admits a finite volume hyperbolic metric (...
6
votes
0
answers
145
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What is the minimum $n$ for which $\Bbb H^3$ can be isometrically embedded in $\Bbb R^n$ as a bounded set?
Consider the hyperbolic $3$-space $\Bbb H^3$ (i.e., the unique, simply-connected, $3$-dimensional complete Riemannian manifold with a constant negative sectional curvature equal to $-1$). The Nash ...
6
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0
answers
237
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Almost parallelizable hyperbolic manifolds
In Sullivan's paper Hyperbolic geometry and homeomorphisms (Proc. Georgia Topology Conf., Athens, Ga., 1977) he makes use of a closed hyperbolic almost parallelizable manifold in every dimension. ($M$ ...
6
votes
0
answers
265
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)}...
6
votes
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80
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"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 ...
6
votes
0
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205
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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
votes
0
answers
202
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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
0
answers
196
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
0
answers
149
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 ...
6
votes
0
answers
383
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
votes
0
answers
199
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
votes
0
answers
388
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
votes
0
answers
159
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Intersection of orbits of earthquake flow on Teichmüller space
Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
5
votes
0
answers
164
views
Counterexamples to the Ahlfors measure conjecture in higher dimensions
Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\...
5
votes
0
answers
148
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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 ...
5
votes
0
answers
72
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 $M$...
5
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0
answers
170
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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 $\geq 3$...
5
votes
0
answers
240
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
votes
0
answers
392
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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 ...
4
votes
0
answers
58
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Existence of finite 3-dimensional hyperbolic balanced geometry
Together with @TarasBanakh we faced the problem described in the title. Let me start with definitions.
A linear space is a pair $(S,\mathcal L)$ consisting of a set $S$ and a family $\mathcal L$ of ...
4
votes
0
answers
93
views
Is there a 5-cell-600-cell honeycomb?
Is there a convex uniform tiling of hyperbolic 4-space with 5-cells and 600-cells as its facets and a snub 24-cell as its vertex figure?
4
votes
0
answers
230
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Convex core and geometric finiteness of negatively curved manifolds
I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...
4
votes
0
answers
132
views
Survey or good reference of taut foliations
I am interested in the topology of foliations.
In particular, I want to understand taut foliations, or projectively Anosov flows, and Anosov flows.
I guess that
A. Candel and L. Conlon, Foliations I (...
4
votes
0
answers
97
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Geodesic foliations of open manifolds foliated by hyperbolic spaces
It is known that hyperbolic spaces admit geodesic foliations (that is, a smooth unit vector field all of whose integral curves are geodesics, see https://arxiv.org/abs/1411.6700).
Suppose a complete ...
4
votes
0
answers
154
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Undecidability for hyperbolic Wang-tilings - pentagons, heptagons, octagons, oh my!
Berger proved that the problem of determining if a finite set of Wang tiles can tile the plane is undecidable. Robinson reproved Berger's result and raised the question of considering the ...
4
votes
0
answers
84
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Correspondence between Hoelder cocycles and Hoelder potential functions for noncompact negatively curved manifolds
Let $\tilde{M}$ be the universal cover of a pinched\ negatively curved manifold $M$ and $\Gamma=\pi_{1}(M)$ its fundamental group and $\partial \Gamma =\partial \tilde{M}$ its Gromov boundary.
When $M$...