All Questions
Tagged with hyperbolic-geometry gt.geometric-topology
264 questions
9
votes
2
answers
2k
views
Translation surfaces
I know that this definitely have some sort of reference out there, but I did not find any wikipidea page for it or any introductory Mathematical article about it . I just want definition and concrete ...
- 1,471
9
votes
1
answer
308
views
Counterexamples to analogue of Cannon conjecture in higher dimensions
It is known that a group $G$ acts geometrically on $\mathbb{H}^2$ if and only if $G$ is word-hyperbolic and its boundary $\partial G$ is homeomorphic to $S^1$.
The analogous statement for $\mathbb{H}^...
- 245
9
votes
2
answers
233
views
Limits at infinity of fellow-travelling sequences in Teichmuller space,
I have a question concerning limits of sequences of points in Teichmuller space, and how this notion is preserved under fellow-travelling.
Suppose that we have closed surface of genus $g\geq 2$, and ...
- 540
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
9
votes
1
answer
518
views
Geodesic cuffs of pairs of pants in a hyperbolic manifold- why are they disjoint?
I'm trying to understand Kahn-Markovic's celebrated Immersing almost geodesic surfaces in a closed hyperbolic three manifold. There is something probably quite basic which I can't figure out.
We have ...
- 22.1k
9
votes
0
answers
308
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 $\rho$.)
...
- 2,533
8
votes
2
answers
642
views
Hyperbolic Brunnian links and rectangular cusp shapes
My question is as follows.
Does every hyperbolic Brunnian link have rectangular cusp shapes on all its components?
Here is what I mean:
The Borromean rings form a famous link $B$ (a smooth closed ...
- 1,131
8
votes
2
answers
375
views
Hyperbolic structures on $S\times\mathbb{R}$
Let $S$ be a closed incompressible surface in a finite volume hyperbolic 3-manifold $M$ without cusps. Let $N$ be the cover of $M$ associated to $\pi_1(S) \subset \pi_1(M)$. The cover $N$ is ...
- 1,601
8
votes
4
answers
601
views
Residual finiteness of hyperbolic 3-manifold groups
So the consequence of the geometrization (according to 3-manifold group note) is that any finite-volumed hyperbolic 3-manifold is residually finite. So the question is:
Q1. If $M$ is an infinite-...
8
votes
1
answer
660
views
On trivial mapping class group of 3-manifolds
What are some examples of knots $K\subset S^3$ such that the mapping class group of $S^3_{1/n}(K)$ is trivial? I guess for hyperbolic knots with no symmetry in the complements are good candidate as ...
- 3,751
8
votes
2
answers
566
views
Pseudo-Anosov maps with same dilatation.
Let $S$ be a hyperbolic surface. Suppose $\mathcal{T}$ denotes the Teichmuller space of $S$ and $Mod(S)$ denotes the mapping class group of $S$. Given any pseudo-Anosov element $f\in Mod(S)$, suppose $...
- 1,713
8
votes
2
answers
667
views
Pseudo-Anosov map, Heegaard splitting, hyperbolic 3-manfold
Hi, I am interested in the relationship between the pseudo-anosov map and volume of the hyperbolic 3-manifold.
Assume $H_{1}$ and $H_{2}$ are two handlebodies with $\partial H_{1}=\partial H_{2}=S$. ...
- 841
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
8
votes
1
answer
613
views
Virtual fibering conjecture for cusped hyperbolic manifolds
I am interested in understanding if the Virtual Fibering Theorem holds in the non-compact case.
Agol proved that every closed hyperbolic $3$-manifold has a finite index cover which fibers over the ...
8
votes
1
answer
352
views
Can I endow the following 3-manifold with a hyperbolic metric?
Consider the following three-dimensional topology. Start with $S^3$ and drill out four unlinked tori as shown in the picture. Then, fill in the gaps with the same tori but with their longitudes and ...
- 355
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
8
votes
1
answer
704
views
Hyperbolic 3-manifolds with no geometrically finite structure
Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer ...
- 29.1k
8
votes
2
answers
631
views
Teichmüller space on non-orientable closed surfaces
It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...
- 81
8
votes
1
answer
611
views
Laminations as a limit of ideal triangulations
I am wondering about the following:
Suppose that $S$ is a non-compact
hyperbolic surface of finite area.
Suppose that $\lambda \subset S$ is a
non-trivial, geodesic,
measured lamination. ...
- 1,329
8
votes
1
answer
292
views
Conditions on the hierarchy for Thurston's hyperbolization theorem
From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, $\partial$-...
- 367
8
votes
1
answer
600
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 ...
- 81
8
votes
0
answers
432
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-...
- 111
7
votes
4
answers
895
views
Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components?
Do you know how to construct a compact hyperbolic 3-manifold with three or four totally geodesic boundary components? The only constructions I could find have one boundary component. A reference ...
- 1,601
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 $G$ ...
7
votes
2
answers
507
views
What is known about exceptional slopes of hyperbolic knots?
For $K$ a hyperbolic knot in $S^3$, a rational number $p/q$ is an exceptional slope if the manifold $M_{p/q}$ obtained from $(p,q)$-Dehn surgery on $K$ does not admit a hyperbolic structure.
Thurston ...
- 2,533
7
votes
2
answers
446
views
Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?
Allcock(2006) proved that
there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$).
His main technique of ...
- 2,581
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
7
votes
1
answer
214
views
Preserving non-conjugacy of loxodromic isometries in a Dehn filling
Suppose that $g$ and $h$ are non-conjugate loxodromic isometries in a cusped hyperbolic $3$-manifold $M$ of finite volume. Fix a cusp $T$ of $M$. Can I choose a hyperbolic Dehn filling of $M$ along $...
7
votes
1
answer
177
views
Ergodicity of action of finite index subgroups in the boundary
Let $\Gamma < \operatorname{PSL}_2(\mathbb{R})= \text{Isom}^+(\mathbb{H^2})$ be a discrete subgroup. Suppose $\Gamma$ acts ergodically on the boundary of the hyperbolic plane $\partial{\mathbb{H}^2}...
- 1,101
7
votes
3
answers
655
views
Hyperbolic 3-manifolds inside algebraic varieties
I have a hyperbolic 3-manifold $M$ that I'd like to see sit inside a complex algebraic variety $V$ as beautifully and snugly as possible. I don't want to specify attributes of "beauty" (e.g.,...
- 121
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
7
votes
2
answers
335
views
Comparing different layered structures for fibered 3-manifolds: example request.
Let's consider a fibering hyperbolic 3-manifold obtained as a mapping torus over some hyperbolic surface with pseudo-Anosov monodromy, and let's suppose that the surface is punctured at the singular ...
- 540
7
votes
1
answer
759
views
Complete geodesics on hyperbolic a pair of pants
I have asked this question on MSE. But I think Mo is a better place to ask my question. Here is the link to my question on MSE. I will rewrite it here:
I am trying to understand the article by Maryam ...
- 231
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
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
7
votes
1
answer
559
views
Standard (special) spines and hyperbolic structure on 3-manifolds
My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
- 367
7
votes
1
answer
215
views
Can the Chern-Simons invariant of a cusped hyperbolic $3$-manfiold be defined mod $\mathbb Z$?
For a closed hyperbolic $3$-manifold $M$, the Chern-Simons invariant $CS(M)$ can be defined as an element of $\mathbb R/\mathbb Z$. When $M$ is cusped it can still be defined, but is now only defined ...
- 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
6
votes
2
answers
889
views
Do quasi convex hyperbolic subgroups remain quasi convex after adding redundant generators?
We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set.
I have been constructing a space ...
- 3,359
6
votes
1
answer
724
views
Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume V?
Is there an explicit bound on the number of tetrahedra needed to triangulate a hyperbolic 3-manifold of volume $V$? Or more generally a hyperbolic $n$-manifold of volume $V$?
- 1,601
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
6
votes
1
answer
2k
views
Definition of cusped manifold?
There is much talk about hyperbolic cusped 3-manifolds, but almost no definition of what a cusped manifold is.
One definition I found was that it is a result of a parabolic transformation on H^n, ...
- 1,465
6
votes
3
answers
647
views
For an arithmetic hyperbolic 3-manifold group, when is its trace field not its invariant trace field?
Edit: In my original post I failed to require the group to be a manifold group. The answer below from @BenLinowitz works in that case. I am really interested though in when the group is torsion-free, ...
- 2,436
6
votes
1
answer
642
views
Andreev's Theorem and Thurston's hyperbolization theorem
I am attempting to get to grips with Thurston's hyperbolization theorem for Haken $3$--manifolds. In particular I was looking at the section related to gluing up along hierarchy surfaces in Otal, Jean-...
- 367
6
votes
1
answer
800
views
Geometrization & JSJ decomposition with boundary
Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
- 125
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
6
votes
1
answer
405
views
Fundamental groups of hyperbolic $4$-manifolds and $\rm CAT(0)$ cube complexes
Suppose $M^4$ is a compact hyperbolic (i.e. curvature $-1$) $4$-manifold and $\Gamma\cong\pi_1(M^4)$.
Is there any expectation whether $\Gamma$ acts properly and co-compactly on a $\rm CAT(0)$ cube ...
- 14.3k
6
votes
2
answers
302
views
Reference for openness of subspace of PSL(2,R) representation variety corresponding to Teichmüller space
$\DeclareMathOperator\PSL{PSL}\DeclareMathOperator\Hom{Hom}$Let $S$ be a compact oriented surface with nonempty boundary. There are two variants of Teichmuller space for $S$ you might consider:
The ...
- 61
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
6
votes
1
answer
297
views
Can bilipschitz models of hyperbolic 3-manifolds be made effective?
In their proof of the Ending Lamination Conjecture, Brock, Canary, and Minsky prove existence of bilipschitz models of hyperbolic 3-manifolds (homeomorphic to a surface times $\mathbb{R}$) depending ...
- 1,601