Questions tagged [hyperbolic-geometry]

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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, ...
Anton Petrunin's user avatar
15 votes
0 answers
887 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 ...
ThiKu's user avatar
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13 votes
0 answers
195 views

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 ...
Ilia Smilga's user avatar
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13 votes
0 answers
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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 ...
David Treumann's user avatar
12 votes
0 answers
418 views

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 ...
BernyPiffaro's user avatar
12 votes
0 answers
454 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\...
Daniil Rudenko's user avatar
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 ...
giulio bullsaver's user avatar
11 votes
0 answers
344 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 ...
SashaKolpakov's user avatar
10 votes
0 answers
369 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,...
Daniil Rudenko's user avatar
10 votes
0 answers
127 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 ...
Ryan Budney's user avatar
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10 votes
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279 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 ...
Dmitri Panov's user avatar
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9 votes
0 answers
235 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 ...
Jugendtraum's user avatar
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, ...
maxematician's user avatar
9 votes
0 answers
346 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 ...
Maik Köster's user avatar
8 votes
0 answers
93 views

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\...
Taras Banakh's user avatar
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8 votes
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242 views

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 $\rho$.) ...
Calvin McPhail-Snyder's user avatar
8 votes
0 answers
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-...
T ghosh's user avatar
  • 81
8 votes
0 answers
920 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 ...
011000's user avatar
  • 81
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 ...
Eric Peterson's user avatar
8 votes
0 answers
189 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 ...
Conifold's user avatar
  • 1,599
7 votes
0 answers
198 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 ...
Mauro's user avatar
  • 191
7 votes
0 answers
331 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 ...
espressionist's user avatar
7 votes
0 answers
1k 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 ...
Clark's user avatar
  • 71
7 votes
0 answers
373 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 ...
strygwyr's user avatar
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6 votes
0 answers
342 views

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 (...
Ian Gershon Teixeira's user avatar
6 votes
0 answers
145 views

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 ...
Random's user avatar
  • 885
6 votes
0 answers
237 views

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. ($M$ ...
Danny Ruberman's user avatar
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)}...
Damaru's user avatar
  • 61
6 votes
0 answers
80 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 ...
user135520's user avatar
6 votes
0 answers
205 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 ...
ThiKu's user avatar
  • 10.2k
6 votes
0 answers
202 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 ...
user17150's user avatar
  • 231
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 ...
Svenja Knopf's user avatar
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 ...
M.U.'s user avatar
  • 701
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 ...
Yellow Pig's user avatar
  • 2,480
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 ...
harlekin's user avatar
  • 313
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 ...
Robert Haraway's user avatar
5 votes
0 answers
159 views

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(\...
Atlas Tasilli's user avatar
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\...
Yankl's user avatar
  • 317
5 votes
0 answers
148 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 ...
Ilya Gekhtman's user avatar
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 $M$...
Moishe Kohan's user avatar
  • 9,624
5 votes
0 answers
170 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 $\geq 3$...
M. Dus's user avatar
  • 1,900
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 ...
seub's user avatar
  • 1,337
5 votes
0 answers
392 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 ...
Arseniy Sheydvasser's user avatar
4 votes
0 answers
58 views

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 ...
Ihromant's user avatar
  • 471
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?
Daniel Sebald's user avatar
4 votes
0 answers
230 views

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 ...
user481559's user avatar
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 (...
user473085's user avatar
4 votes
0 answers
97 views

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 ...
Claudio Gorodski's user avatar
4 votes
0 answers
154 views

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 ...
user101010's user avatar
  • 5,319
4 votes
0 answers
84 views

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 $M$...
Ilya Gekhtman's user avatar

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