All Questions
Tagged with hyperbolic-geometry gt.geometric-topology
264 questions
10
votes
2
answers
488
views
Question about the Weeks Manifold
I am wondering about the existence of incompressible surfaces in the Weeks manifold. Is this space a Haken manifold?
- 3,165
3
votes
1
answer
375
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Chern-Simons invariants of 2-bridge knots
2-bridge links $L(p/q)$ are described by the continuous fraction expansion $\frac{p}{q}=\left[a_1,a_2,\ldots,a_n\right]$, where the $a_i$ are the numbers of twists in the boxes below:
Looking at some ...
CommunityBot
- 1
1
vote
2
answers
310
views
Are there CAT(-1) spaces which are not trees whose Gromov boundary is disconnected?
Are there some examples of CAT(-1) spaces which are not trees which have disconnected Gromov boundary?
- 25k
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 ...
- 15.4k
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 ...
- 63.9k
9
votes
2
answers
812
views
How many knots are there with hyperbolic volume less than a given constant
Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...
CommunityBot
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2
votes
0
answers
212
views
Exotic actions of hyperbolic groups
Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that:
The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$.
For every quasiconvex subgroup $H \leq G$...
- 11.3k
9
votes
4
answers
1k
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Geometry of the space of circles in the Euclidean plane
We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.
It may even be possible to write an explicit formula ...
- 437
14
votes
2
answers
617
views
For which surfaces is Penner's conjecture known to be true?
Robert Penner has proven that, if $A=\{a_1,\dots, a_n\}$ and $B=\{b_1,\dots, b_m\}$ are multicurves in a surface $S$ that together fill $S$, then any product of positive powers of Dehn twists along ...
- 10.6k
10
votes
2
answers
650
views
Heegaard genera of arithmetic 3-manifolds
UPDATE: Because I was hoping that state the question as concisely as
possible, the original post did not include a precise definition of
arithmetic 3-manifold only a reference to Maclachlan and ...
- 5,259
1
vote
1
answer
271
views
Ratner theorem and dense geodesic planes in hyperbolic manifolds
Suppose we have a closed hyperbolic $3$-manifold $M$. For any $x\in M$ and plane $\pi$ in $T_xM$ we consider $P$ the geodesic plane exp$(\pi)$ originating from $\pi$. For any $p\in \pi$ we consider ...
- 17k
1
vote
1
answer
130
views
Structures on open surfaces
Let $\phi\in PSL(2,R)$ be hyperbolic and $\varphi\in PSL(2,R)$ be elliptic.
Is it possible to find a local homeomorphism $f:H^2\rightarrow H^2$ such that
$f(\phi(x))=\varphi(f(x))$ for all $x\in H^2$ ?...
- 31.2k
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 ...
- 5,259
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 ...
- 25k
11
votes
1
answer
332
views
Rank and hyperbolic volume
Suppose $M$ is a hyperbolic $3$-manifold whose fundamental group has rank $r.$ What is the best (lower) bound on the volume of $M?$ Similar question for rank of $H_1.$ There is a bunch of papers of ...
- 5,382
9
votes
3
answers
801
views
What constant ensures hyperbolicity of Dehn surgery?
I am interested in showing that certain knots having a surgery description are hyperbolic. Unfortunately I have not had time yet to read Thurston's work, so my understanding of this is vague. But from ...
- 5,259
11
votes
2
answers
620
views
Some mid-sized ¿hyperbolic? manifolds and SnapPea
I've recently come across some mid-sized 3-manifolds that I think are likely hyperbolic, but SnapPea has some trouble with them. This is related to my previous question
can you fool SnapPea?
but in ...
CommunityBot
- 1
5
votes
1
answer
629
views
What is known about maximal free subgroups of surface groups?
Let $\Gamma_g=< a_1,...,a_g,b_1,...,b_g | \prod_{i=1}^g [a_i,b_i]>$ (a surface group). What is known about maximal free subgroups of $\Gamma_g$ for $g>1.$ (I.e. free subgroups which are not ...
- 68.9k
3
votes
1
answer
227
views
Density of ends of long words in a hyperbolic group
Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...
- 25k
5
votes
1
answer
193
views
Injective simplicial maps between Arc complexes
Let $A(S)$ denotes the Arc complex of a finite type hyperbolic surface $S$ with nonempty boundary. Let $\lambda:A(S)\rightarrow A(S)$ be a map such that on triangulations of $S$ i.e. on the top ...
- 15.5k
11
votes
0
answers
352
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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 ...
CommunityBot
- 1
18
votes
3
answers
1k
views
The number of cusps of higher-dimensional hyperbolic manifolds
Suppose $n$ is an integer greater than 3. Sometimes ago I heard somewhere that it is still not known if there exist complete finite-volume hyperbolic $n$-manifolds having exactly one cusp.
Could ...
- 1,268
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) ...
- 10.4k
0
votes
1
answer
291
views
A question on Cayley graphs and hyperbolic 3-manifolds
There are two hyperbolic closed 3-manifolds, but I don't know whether they are homeomorphic or not. The only thing I know is that the Cayley graphs of their fundamental groups are quasi-isometric.
...
- 28.2k
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
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
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-...
- 38.6k
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 ...
CommunityBot
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2
votes
0
answers
199
views
Discrete Isoperimetric Problem in the Hyperbolic Plane
Let $S_{g}$ be the closed orientable genus $g$ surface. I'm interested in studying a certain class $\Gamma$ of graphs on $S_{g}$ which fill (the complementary regions are simply connected), and which ...
- 198
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 ...
- 1,329
4
votes
1
answer
1k
views
Hyperbolic structures on once punctured tori
I've been working on a problem about billiards in ideal hyperbolic polygons and I was thinking about how the problem for ideal quadrilaterals relates to closed geodesics on once punctured tori.
My ...
- 15.4k
3
votes
2
answers
1k
views
Hyperbolic pair of pants.
Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential simple arc $\sigma$ ...
- 1,106
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 ...
- 15.4k
11
votes
1
answer
902
views
Why isn't $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}\rangle$ a hyperbolic surface group?
The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed torus fundamental ...
- 15.4k
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. ...
- 5,259
0
votes
2
answers
863
views
A quick question about Farb-Margalit's book on MCG's proof on Teichmuller's existence theorem
Hello,
I was studying Farb-Margalit's " A Primer on MCG " for Teichmuller's existence theorem. On P. 347, proposition 11.14, they proved $ \omega : QD_1(X) -> Teich ( S_g) $ is proper, which, ...
- 1,713
12
votes
2
answers
2k
views
The work of Thurston
I seem to remember written or said somewhere that at some point Thurston decided to stop writing down his theorems in order not to repel mathematicians from his field (maybe this is not correct?). I ...
- 8,842
18
votes
2
answers
2k
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How does hyperbolicity of space time affect our lives?
My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold.
But recently I found ...
- 199
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 ...
- 28.2k
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 ...
- 68.9k
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 ...
- 10.4k
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 ...
- 25k
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 $\partial H_{1}=\partial H_{2}=S$. ...
CommunityBot
- 1
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 ...
- 15.4k
0
votes
0
answers
119
views
uniform properness of lifts of uniform proper maps
Let $(X,d)$ be a unique geodesic metric space and $Y\subseteq X$ a subset, equipped with the induced path metric $d'$. We say that the inclusion $i\colon Y\hookrightarrow X$ is uniformly proper if $\...
- 213
5
votes
3
answers
1k
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Matrices generating non-discrete subgroups of SL(2,R)
Jorgensen's inequality $\mid \left(Tr\left(A\right)\right)^2-4\mid+\mid Tr\left[A,B\right]-2\mid\ge 1$ gives a necessary condition for two matrices A,B to generate a discrete subgroup of SL(2,R). Are ...
- 31.2k
5
votes
1
answer
464
views
Does the fundamental group of a surface have rigid subgroups?
Suppose $\Gamma$ is the fundamental group of a closed, oriented surface $S$. Let $B$ be a finitely generated, infinite index subgroup of $\Gamma$, and let $\Gamma_B$ be the compact core of the $B$-...
CommunityBot
- 1
10
votes
1
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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 ...
- 31.2k
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 $G$?
- 31.2k