Questions tagged [hyperbolic-geometry]
865 questions
9
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Questions on Thurston's earthquake flow
Here are some questions about the earthquake deformation of hyperbolic surface that I can't answer or find references.
I briefly recall the settings. Let's fix a closed surface $S$ with genus $g\geq 2$. A point $h$ in the Teichmuller space…
Xin Nie
- 1,764
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9
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4 answers
Canonical fundamental domain for a discrete subgroup Γ of SL₂(R) acting on hyperbolic plane
Let a discrete subgroup $\Gamma$ of $SL_2(\mathbb R)$ act on the hyperbolic plane by Möbius transformations. Is there a "best" or "most canonical" fundamental domain for this action?
Some (mostly unhelpful) observations:
For the action of…
Akela
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9
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4 answers
It is well-known that hyperbolic space is delta-hyperbolic, but what is delta?
Recall that a space is $\delta$-hyperbolic if there is some number $\delta$ with the property that every point on an edge of a geodesic triangle lies within $\delta$ of another edge. For example a tree is $0$-hyperbolic. One of the basic facts…
Paul Siegel
- 28,772
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Does every hyperbolic 3 manifold with totally geodesic boundary has some finite covering space with more than one boundary component?
I am thinking about the question that: if we double a hyperbolic 3 manifold along its boundary, will the rank of fundemental group of the resulting closed manifold be strictly larger than before?\
The answer is "Yes" if the following question is…
strygwyr
- 205
7
votes
2 answers
Poincaré disk model: is this locus a known curve?
Please, consider a line segment $AB$ in the Poincaré disk model. Now, consider the set $S$ of all point $P$ in the disk such that the angle $\angle APB$ is constant.
Question: is $S$ a known curve?
Thanks!
7
votes
1 answer
Distances between boundaries in a hyperbolic pants
Consider a hyperbolic pair of pants with totally-geodesic boundaries of lengths $l_i$ for $i \in \{1,2,3\}$. For any two distinct boundary components, is the length of the shortest geodesic connecting them already determined? If so, is there a…
Jamie Vicary
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6
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A question about embedding hyperbolic space onto pseudosphere
I have a difficulty with hyperbolic geometry.
Let $\mathbb{H}^{2}$ be a 2-dimensional hyperbolic plane.
(i.e., upper half plane in $\mathbb{R}^{2}$ with a metric $\frac{ds}{y}$)
(or, upper half plane in $\mathbb{C}$ with a metric…
KENSO
- 137
6
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1 answer
Can most 3 dimensional hyperbolic orbifolds with finite volume be covered by a hyperbolic manifold?
If G is a discrete cofinite volume subgroup of PSL(2,C),then G acts on H3, H3/G is a 3-dim hyperbolic orbifold N with finite volume, my question is : Is it right in most situations that we can find a hyperbolic 3 manifold M as a finite covering…
strygwyr
- 205
5
votes
3 answers
center of fundamental group of finite volume-hyperbolic orbifold
Is center of a fundamental group of finite volume-hyperbolic n-orbifold trivial?
Is there a good reference that the proof is wriiten?
user9552
- 149
5
votes
1 answer
The geometrical meaning of the common value in the law of sines in hyperbolic geometry
What is the geometrical meaning of the common value in the law of sines, $\frac{\sin A}{\sinh a} = \frac{\sin B}{\sinh b} = \frac{\sin C}{\sinh c}$ in hyperbolic geometry? I know the meaning of this value only in Euclidean and spherical geometry.…
zar
- 151
5
votes
1 answer
How people think about ending lamination?
There are lots of works in hyperbolic 3-manifolds related to ending lamination.
But I just don't know how people think about it . what is the philosophy behind it?
maybe i should ask how people thought to use lamnination to classifying 3-hyperbolic…
yanqing
- 146
5
votes
0 answers
Covering hyperbolic manifolds by round balls
This question is a natural follow up to the question asked here. I think it should not be too hard to answer it (negatively) in dimension 3, but higher dimensions will be probably challenging.
Let $M$ be a connected hyperbolic $n$-manifold. A round…
Moishe Kohan
- 9,664
5
votes
2 answers
Area of hyperbolic triangle in terms of Lengths of its sides
Let a, b and c denote the cosh of the lengths of the
sides of an hyperbolic triangle and A, B, and C its angles.
Its area is well knwon to be S = pi - A - B - C .
What is S in terms of a, b, c ?
In J. Smorodinskij, Fortschritte der Physik, 18 (1965)…
Norbert Dragon
- 155
4
votes
1 answer
Pleated surfaces do not curl up too much
Hi!
Let $S$ be a hyperbolic surface with metric $\rho$ and $N$ a hyperbolic $3$-dimensional manifold with bounded geometry. Let $g\colon (S,\rho)\to N$ be an incompressible pleated surface, that is to say:
$g$ is a path-isometry (it maps paths of…
Damiano Lupi
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4
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Coordinates on Teichmuller space
We know that every surface of genus ($g\geq 2$) admits a pair of pants decomposition. And there is the Fenchel Nielsen Coordinates on the Teichmuller space associated to such a decomposition where we have the length functions of the geodesics…
Junwu Tu
- 105