All Questions
73 questions
4
votes
0
answers
302
views
Haken manifolds and characterising sutured manifold hierarchies
In Gabai's paper (Knot Theory and Manifolds Lecture Notes in Mathematics Volume 1144, 1985, pp 14-17 An internal hierarchy for 3-manifolds) he considers sutured manifold decompositions of Haken $3$--...
- 367
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
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
9
votes
2
answers
691
views
Do different Dehn fillings produce homeomorphic 3-manifolds ?
Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold.
Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a
hyperbolic structure. ...
- 841
6
votes
0
answers
389
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 ...
- 1,131
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
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
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
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
10
votes
1
answer
501
views
Examples of 3-manifolds with RFRS fundamental group
I'm wondering if anyone knows how to construct hyperbolic 3-manifolds whose fundamental group is RFRS. Clearly the recent work of Agol, Wise, etc. says that such manifolds are abundant, and in ...
- 1,329
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
13
votes
3
answers
1k
views
Best known Margulis constants?
A Margulis number for a hyperbolic $n$-manifold $M=\mathbb{H}^n/\Gamma$ is a number $\epsilon>0$ such that for each $x\in\mathbb{H}^n$ the group generated by the elements in $\Gamma$ which move $x$ ...
- 1,601
4
votes
1
answer
199
views
Rank of a group generated by side-pairing isometries of a polyhedron
Let $P$ be a compact convex polyhedron in $\mathbb{H}^3$. Let $G$ be a group generated by side-pairing isometries of $P$. Is there an algorithm to find the rank of $G$?
- 1,601
4
votes
1
answer
238
views
Mapping torus relative to an infinite orbit can be hyperbolic with finite volume?
Consider a homeomorphism of the sphere with an infinite orbit converging forwards and backwards to the same point. Remove the orbit and the accumulation point and make the mapping torus. Can the ...
18
votes
3
answers
1k
views
How to see isometries of figure 8 knot complement
The figure 8 knot complement $M$ is the orientable double cover of the Gieseking manifold, which implies that $M$ has a fixed-point free involution. If we think of $M$ with its hyperbolic metric, this ...
- 518
13
votes
1
answer
2k
views
Detecting a cover of the figure-8 knot complement
I have a specific concrete example of a complete, finite volume, cusped hyperbolic 3-manifold $M$, and I am trying to determine whether or not $M$ is a cover of the figure-8 knot complement (call this ...
- 418
12
votes
3
answers
1k
views
Is the cut locus of a generic point in a hyperbolic manifold a generic polyhedron?
Let $p\in M$ be a point in a closed riemannian manifold $M$. Recall that the cut locus of $p$ is the subset of $M$ consisting of all points that are connected to $p$ by at least 2 distance-minimizing ...
- 10.5k
10
votes
1
answer
568
views
Is the spectrum of closed geodesics in a closed hyperbolic 3-manifold asymptotically homogeneously dense?
It seems to me that the results in this important paper of Kahn-Markovich imply the following fact. Let $M^3$ be any closed hyperbolic 3-manifold. For every $\epsilon > 0$ there is a natural number ...
- 10.5k
1
vote
1
answer
484
views
Why do strongly irreducible Heegaard surfaces look like fibers?
I remember hearing somewhere that strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds "look like" fibers.
I know that Otal's result about short geodesics in hyperbolic mapping tori being ...
- 1,601
5
votes
1
answer
439
views
Rotation part of short geodesics in hyperbolic mapping tori
Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
- 1,601
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
13
votes
3
answers
1k
views
Flat SU(2) bundles over hyperbolic 3-manifolds
Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold?
The literature on such bundles over 3-manifolds is huge and my naive searches don'...
- 6,247
5
votes
2
answers
515
views
Contracting maps of hyperbolic manifolds
Is it possible for SOME positive $c$, $c<1$ to find a pair of COMPACT hyperbolic manfiolds $M^3$ and $N^3$
with a positive degree map $$f: M^3 \to N^3,$$ such that $f$ is contacting with constant $...
- 28.9k