All Questions
Tagged with hyperbolic-geometry gt.geometric-topology
264 questions
1
vote
2
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925
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Length of shortest geodesic and Cheeger's isoperimetric constant for a special genus 2 surface
Let us take two copies of $ Y $-pieces [ or pair of pants ] with each boundary geodesic of length $ l $, and glue them together without any twisting to obtain a genus 2 closed orientable hyperbolic ...
- 1,471
12
votes
3
answers
1k
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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
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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
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1
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484
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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
6
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1
answer
724
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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
18
votes
3
answers
1k
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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 ...
- 3,749
5
votes
1
answer
439
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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
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4
answers
895
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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
19
votes
1
answer
901
views
Locus of equal area hyperbolic triangles
Henry Segerman and I recently considered the following question:
Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
- 565
13
votes
2
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1k
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Closed hyperbolic manifold with right-angled fundamental domain
What is an example (as simple as possible, please!) of a closed hyperbolic three-manifold with a right-angled polyhedron as fundamental domain?
If we allow cusps then the Whitehead link or the ...
- 28.2k
3
votes
0
answers
173
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Collapsing the medial axis of a polytope
Let X be a convex polyhedron in hyperbolic 3-space.
Let M be the medial axis of X.
Question: Is M collapsible?
It is easy to see that M is contractable.
In the case of Euclidian 3-space, instead ...
- 133
13
votes
3
answers
1k
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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
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 ...
- 28.9k