All Questions
Tagged with hyperbolic-geometry gt.geometric-topology
264 questions
8
votes
1
answer
572
views
Mostow Rigidity Theorem and reconstruction from fundamental group
The Mostow Rigidity Theorem is phrased in terms of a relationship between isometries and isomorphisms of fundamental groups, which raises an obvious question. Given the fundamental group of a complete ...
- 3,816
5
votes
1
answer
248
views
Explicit check of the invariance of the Weil-Petersson form
Using Fenchel-Nielsen coordinates, the Weil-Petersson metric can be written as
$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$
where $i$ is an index labelling the curves of a pants decomposition of ...
- 1,435
4
votes
2
answers
485
views
When existence of loxodromic, WPD elements implies an action is acylindrical
Definitions
Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries.
Recall that an action of $G$ on $X$ is called acylindrical if the ...
- 421
3
votes
1
answer
853
views
Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures
In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...
- 1,435
6
votes
1
answer
466
views
Hyperbolic manifolds with infinite cyclic fundamental group
It is a well known fact that there is a correspondence between complete hyperbolic $n$-manifolds up to isometry and discrete subgroups of isometries of the hyperbolic space $\mathbb{H}^n$ that act ...
- 2,533
7
votes
1
answer
372
views
Thickness and hierarchical hyperbolicity
Thick metric spaces were introduced by Behrstock, Drutu and Mosher, see here. Hierarchically hyperbolic spaces were introduced by Behrstock, Hagen and Sisto, see here.
I've heard that it is open ...
- 2,090
4
votes
1
answer
392
views
Topology on the boundary compactification $X^{-}=\partial X\cup X$ of a Gromov-hyperbolic space
Consider a proper geodesic $\delta$-hyperbolic space $X$ (in the sense of Gromov). Let ∂𝑋 be its Gromov boundary. In the book "Geometric Group Theory" by Cornelia Druţu and Michael Kapovich https://...
- 313
5
votes
1
answer
342
views
Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...
- 2,090
4
votes
1
answer
2k
views
On Thurston's triangulations of sphere
I have two questions from Thurston's paper [1].
In the paper [1], Thurston talks about classifying certain classes of triangulations of the sphere. Here a triangulation of a sphere a Topological ...
- 239
6
votes
1
answer
189
views
Starting letters of equivalent infinite geodesic paths of hyperbolic Coxeter groups
Let $\left(W\text{, }S\right)$ be a Gromov hyperbolic Coxeter system and denote by $\partial W$ the corresponding Gromov boundary. For $z\in\partial W$ let $\alpha$, $\beta$ be infinite geodesic paths ...
- 741
3
votes
0
answers
267
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,...
- 679
5
votes
0
answers
183
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 $\geq 3$...
- 2,090
1
vote
0
answers
202
views
Quadrilateral fundamental domain
Let P be a hyperbolic quadrilateral. Poincare polygon theorem provides sufficient condition for P to be a fundamental domain of some Fuchsian group in term of its inner angles. I find the angle sum ...
- 745
4
votes
2
answers
493
views
Tiling of genus 2 surface by 8 pentagons
In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons".
Question 1: What does this tiling look like?
Question 2:...
- 43
7
votes
1
answer
204
views
A coincidence between the Lambert cube, Lobell polyhedron, and hyperbolic 3-manifolds?
I. Lambert cube $\mathfrak L(\alpha_1,\alpha_2,\alpha_3)$
In this paper (p.8), we find the volume $V$ of the hyperbolic Lambert cube for the special case $\alpha=\alpha_1 = \alpha_2 = \alpha_3$ as
$...
- 12.6k
5
votes
1
answer
214
views
Asking SnapPy for core curves after surgery
Suppose I give SnapPy a cusped hyperbolic 3-manifold (using, say, the link editor) and specify some filling. SnapPy can then provide a presentation of the fundamental group of the filled manifold. Can ...
- 515
2
votes
2
answers
330
views
$PSL_2(\mathbb{R})$ representations of free groups
Let $S_{g,n}^b$ denote a surface of genus $g$ with $n$ punctures and $b$ boundary components. Let us assume $\max\{b,n\}\geq 1$. It is then obvious that $S_{g,n}^b$ deformation retracts to a bouquet ...
- 445
3
votes
1
answer
569
views
Hyperbolic 3 manifold with trivial deformation of flat conformal structures
Does there exist closed hyperbolic three manifold which is locally rigid in the space of its all conformally flat structures? If so could someone provide examples?
- 171
7
votes
1
answer
546
views
Can a hyperbolic manifold be a product?
I was interested in whether a manifold which admits a metric of constant sectional curvature can be homotopy equivalent to a product of non-contractible manifolds. Of course, there are three cases: ...
- 19.3k
6
votes
0
answers
81
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 ...
- 401
9
votes
2
answers
886
views
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 ...
- 29.1k
1
vote
0
answers
218
views
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 ...
- 21
13
votes
2
answers
485
views
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 ...
- 10.1k
3
votes
0
answers
110
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 ...
user127549
7
votes
4
answers
454
views
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 $G$ ...
6
votes
1
answer
575
views
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 ...
user74900
40
votes
1
answer
1k
views
Four circles on the sphere
Consider generic 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 ...
- 91.8k
5
votes
1
answer
292
views
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 ...
- 309
5
votes
1
answer
245
views
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?
- 553
4
votes
1
answer
317
views
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 ...
- 553
10
votes
1
answer
738
views
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 ...
- 2,090
1
vote
0
answers
61
views
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 ...
- 1,435
5
votes
1
answer
142
views
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 ...
- 14.3k
5
votes
1
answer
394
views
closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface
Could you please recommend me some references for proofs of this fact: "closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface". Thanks in advance!
- 333
5
votes
2
answers
408
views
Smallest tile to *isohedrally* tessellate the hyperbolic plane
Is there a smallest tile (in terms of diameter) that isohedrally tessellates the hyperbolic plane?
In this question, we ask the same question without the isohedral requirement, and the answer was no. ...
- 6,353
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 ...
user50806
20
votes
2
answers
2k
views
Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
- 6,353
3
votes
0
answers
196
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 ...
user50806
7
votes
1
answer
475
views
Can we cut and rotate a particular region of a hyperbolic 3-manifold to get another (non-homeomorphic) hyperbolic 3-manifold?
I'm trying to learn more about hyperbolic 3-manifolds, in particular the geometric implications of doing hyperbolic Dehn surgery to suitable knot complements.
Following this paper by Christian ...
- 318
3
votes
1
answer
496
views
Classifying links with essential annuli in the complement as torus links
I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and ...
- 33
3
votes
0
answers
232
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 ...
- 2,436
4
votes
1
answer
249
views
Trace field of a hyperbolic $3$-manifold with a totally geodesic subsurface
Let $X$
be a complete finite-volume orientable hyperbolic $3$-manifold,
and let $\Gamma$
be a Kleinian representation of $\pi_1(X)$.
Let $K\Gamma:=\mathbb{Q}\big(\{\mathrm{tr}\mid\gamma\in\Gamma\}\big)...
- 2,436
6
votes
3
answers
555
views
Given a link $L\subset S^3$ how to construct a link $L'$ whose complement have hyperbolic structure?
Thurston claimed that almost all closed 3-manifolds are hyperbolic. To support this, he said that every closed 3-manifold is obtained by Dehn surgery along some link whose complement is hyperbolic. ...
- 3,751
9
votes
2
answers
385
views
Are pseudo-Anosov foliations dense?
A pseudo-Anosov foliation of a compact orientable surface $F$ is a one whose class in the space $\mathcal{PMF}(F)$ of projective measured foliations is preserved by some pseudo-Anosov homeomorphism of ...
- 2,390
8
votes
2
answers
793
views
Does there exist a Haken manifold where all its incompressible surfaces are non-separating?
Every non-zero element in $H_2(M,\mathbb Z)$ corresponds to an incompressible surface. So these surfaces are non-separating. But I'm interested in knowing about separating incompressible surfaces. A ...
- 3,751
18
votes
2
answers
1k
views
Hyperbolic Volume and Chern-Simons
In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons ...
- 183
5
votes
1
answer
307
views
Does there exist a closed geodesic go through a $\epsilon$-net of a hyperbolic surface?
An $\epsilon$-net of a closed hyperbolic surface $X$ is a finite set of points $p_i$ such that the family of balls centered at $p_i$ with radius $\epsilon$ is a cover of $X$, and the family of balls ...
- 333
3
votes
2
answers
950
views
hyperbolic 3-manifold of finite volume
Is there a complete description of hyperbolic 3-manifold of finite volume ?
Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume?
...
- 1,629
7
votes
1
answer
700
views
Bers' constant for compact hyperbolic surfaces with geodesic boundary
The clasical Bers' theorem about pants decomposition says that any compact Riemann surface of genus $g \geq 2$ has a pants decomposition such that every cutting geodesic in this decomposition is of ...
- 305
4
votes
0
answers
205
views
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 ...
- 2,090