Questions tagged [hyperbolic-geometry]
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81 questions
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The Teichmüller space $T_g$ of a closed riemann surface $S_g$ of genus $g \geq 2$ can't be parametrized by $6g−6$ geodesic length functions
I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty ...
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votes
3
answers
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Thurston's 24 questions: All settled?
Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":
$\cdots$
Two naive questions from an outsider:
(1) Have all $24$ now been resolved?
(2)...
- 151k
27
votes
2
answers
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Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds
Thurston's Hyperbolic Dehn Surgery Theorem says that all but finitely many fillings of a cusp of a hyperbolic 3-manifold result in hyperbolic manifolds that are deformations of the original manifold. ...
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5
answers
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can you fool SnapPea?
A while back I thought I had some simple knots that "fooled" SnapPea. But I no longer remember those examples, if I ever had them to begin with.
What I'm looking for is a non-hyperbolizable knot ...
- 44.4k
22
votes
2
answers
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Non-residually finite matrix groups
By Malcev's theorem, every finitely generated linear group is residually finite (RF).
On the other hand, say, the group of rational numbers is linear, but is not residually finite. Thus, one has to ...
- 31.2k
17
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2
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2k
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Geodesics on the twisted pseudosphere (Dini's surface)
I wonder how difficult it is to compute geodesics on Dini's Surface,
a twisted pseudosphere?
Here is one parametrization, from
Alfred Gray's Modern Differential Geometry of Curves and Surfaces, p....
- 151k
14
votes
1
answer
852
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Cutting up the Bring surface into six pairs of pants
The Bring sextic, with 120 automorphisms, is the numerically most symmetric compact Riemann surface of genus 4. To cut it up into six pairs of pants, we need to cut along nine disjoint geodesic loops....
- 423
12
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2
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Existence of finite index torsion-free subgroups of hyperbolic groups
Question. Is it true that each infinite hyperbolic group
has a torsion-free subgroup of finite index?
Are there counterexamples, or positive results for some large subclasses of hyperbolic groups?
For ...
- 28.9k
10
votes
4
answers
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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 ...
- 29.2k
9
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1
answer
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Change of coordinates for Teichmüller space of the 4-holed sphere
The diagram below indicates 2 ways to use Fenchel-Nielsen coordinates to parameterize the Teichmüller space of conformal structures on the 4-holed sphere with totally-geodesic boundary, corresponding ...
- 2,513
9
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1
answer
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Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?
In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
- 1,731
9
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1
answer
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How did Gauss characterize the metrical relations in the uniform (4 4 4) tiling of the hyperbolic unit disk?
My purpose is to verify an historical hypothesis I have on Gauss's tesselation of the unit disk as described in John Stilwell "Mathematics and its history". Looking at the relevant pages in ...
- 2,099
31
votes
4
answers
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Why do people study representations of 3-manifold groups into $SL(n,\mathbb{C})$?
Varieties of representations and characters of $3$-manifold groups in $SL(2,\mathbb{C})$ have been intensively studied. They have provided tools to identify geometric structures on manifolds, and are ...
- 22.1k
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votes
7
answers
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Tetrahedra with prescribed face angles
I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is spherical/euclidean/...
- 11.1k
26
votes
4
answers
1k
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Equations defining hyperbolic geodesics in $\mathbb C \setminus\{0,1\}$
Let $X=\mathbb C\setminus\{0,1\}$, equipped with the hyperbolic structure it inherits from Klein's modular $\lambda$ function $\lambda:\mathbb H \to X$. In each (non-peripheral and nontrivial) free-...
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24
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4
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Immersions of the hyperbolic plane
Is it possible to isometrically immerse the hyperbolic plane into a compact Riemannian manifold as a totally geodesic submanifold? Any nice examples?
Edit: Although I did not originally say so, I was ...
- 13.5k
21
votes
1
answer
753
views
Complexifications of hyperbolic manifolds
I'm wondering when a compact hyperbolic $n$-manifold ($n \geq 3$) can embed in a complex hyperbolic $n$-manifold as a real algebraic subvariety so that it is a component of the fixed point set of ...
- 68.9k
20
votes
2
answers
2k
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Smallest tile to tessellate the hyperbolic plane
Is it known what the smallest tile (in terms of area) that can tessellate the hyperbolic plane is? In particular, it should tessellate the plane by itself.
I think it will be a Triangle group, but I'...
- 6,353
20
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1
answer
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Canonical immersion of the double torus
It is easy to check that the immersion $\mathbb{T}^2=\mathbb{S}^1\times \mathbb{S}^1\longrightarrow\mathbb{R}^4$, $(\alpha,\beta)\longmapsto(\cos\alpha,\sin\alpha,\cos\beta,\sin\beta)$ induces the ...
- 2,091
18
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2
answers
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Reference request: Geodesic flow on a manifold with negative curvature is ergodic
I'm reading about the Mostow's rigidity theorem, and the proof uses the following (maybe well-known) result:
The geodesic flow on a manifold with negative curvature is ergodic.
The lecture note that ...
- 957
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
17
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3
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Examples of the Thurston geometries with transitive Lie group action
Here are some examples of compact homogeneous 3 manifolds for different Thurston geometries:
(1) Spherical: $\mathbb{S}^3 \cong \mathrm{SU}_2$ modulo any finite subgroup
(2) Euclidean: 3 torus $\...
- 4,081
16
votes
2
answers
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The fundamental group of a closed surface without classification of surfaces?
The fundamental group of a closed oriented surface of genus $g$ has the well-known presentation
$$
\langle x_1,\ldots, x_g,y_1,\ldots ,y_g\vert \prod_{i=1}^{g} [x_i,y_i]\rangle.
$$
The proof I know ...
- 20.9k
16
votes
2
answers
1k
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Maximum of a function of one variable
Let $D$ be a circular quadrilateral (that is a Jordan region whose boundary consists of 4 arcs
of circles all orthogonal to the unit circle) whose interior angles are all equal to 0, the vertices lie ...
- 91.8k
15
votes
2
answers
2k
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Dehn's algorithm for word problem for surface groups
For some $g \geq 2$, let $\Gamma_g$ be the fundamental group of a closed genus $g$ surface and let $S_g=\{a_1,b_1,\ldots,a_g,b_g\}$ be the usual generating set for $\Gamma_g$ satisfying the surface ...
- 153
15
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0
answers
927
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How does duality of symmetric spaces explain the hyperbolic cosine theorem?
There is a well-known duality between compact symmetric spaces and symmetric spaces of noncompact type. Basically it goes as follows: If $$G/K$$ is a symmetric space of noncompact type, $$g=k+p$$ the ...
- 10.4k
15
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1
answer
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Pythagorean theorem for right-corner hyperbolic simplices?
My answer to the "Favorite equations" question was the Pythagorean theorem for right-corner tetrahedra:
Euclidean: $A^2+B^2+C^2=D^2$
Hyperbolic: $\cos\frac{A}{2}\cos\frac{B}{2}\cos\frac{C}{2}−\...
- 1,230
13
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3
answers
2k
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Isometry group of a compact hyperbolic surface
Consider a compact surface $M$ of genus $g \geq 2$ with a metric of constant negative curvature. My question is, is it known under what sorts of sufficient conditions such a metric will have non-...
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13
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1
answer
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Topology of boundaries of hyperbolic groups
For many examples of word-hyperbolic groups which I have seen in the context of low-dimensional topology, the ideal boundary is either homeomorphic to a n-sphere for some n or a Cantor set. So, I was ...
- 305
13
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3
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851
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Are negatively pinched manifold locally conformally flat?
One knows that hyperbolic manifolds are locally conformally flat.
How about those negatively pinched manifolds, i.e. the sectional curvature $K$ satisfy:
$$
-\Lambda \le K \le -\lambda$$
for $\Lambda&...
- 2,623
12
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3
answers
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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
11
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3
answers
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Random links and $3$-manifolds
In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:
A non-trivial connected sum $M_1\# M_2$ ...
- 1,314
11
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4
answers
1k
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Surfaces with non-constant negative curvature
Are there any nice models of surfaces with non-constant negative curvature, analogous to the Poincare disk for constant negative curvature. I have found lots of general results and theory but no nice ...
- 680
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2
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Can you cover a genus a billion hyperbolic surface with 15 balls?
Here's a question I was wondering about this week. Not sure how interesting it is, but I thought it was kind of curious.
Question: Given $k$, is there a number $N=N(k)$ such that if a closed ...
- 532
11
votes
3
answers
748
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Explicit triples of isomorphic Riemann surfaces
Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....
11
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4
answers
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Geodesics on a hyperbolic paraboloid
Given any two points on a hyperbolic paraboloid ($xy = z$ or $z = (x^2 - y^2)/2$) how does one find the geodesic between them?
I know that since the hyperbolic paraboloid is doubly ruled, some of the ...
- 529
10
votes
1
answer
579
views
Periodic billiard paths in hyperbolic triangles
It is a theorem of Masur that all rational triangles in the Euclidean plane posses a periodic billiard path,
one obeying the reflection law that the angle of incidence equals the angle of reflection. ...
- 151k
10
votes
1
answer
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fundamental domains for free fuchsian group.
I try to understand some of the topology of the space of pointed non-compact hyperbolic surfaces (with the pointed Gromov-Hausdorff topology). It is known that the fundamental
group of a non-compact ...
- 14.4k
10
votes
3
answers
602
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Torsion in cuspidal cohomology
Following Lemma 2.7 from Vogtmann's Rational Homology of Bianchi Groups, I want to define cuspidal cohomology as $$H_{\mathrm{cusp}}(M)=\frac{H_1(M)}{i_*(H_1(\partial M))}$$ where $i:\partial M\to M$ ...
- 656
9
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1
answer
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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}^...
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Fundamental group of an hyperbolic $4$-manifold
Good afternoon everyone,
I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension (at least $4$). If the theory of surfaces and $3$-manifolds provide ...
- 2,696
9
votes
4
answers
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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 ...
- 3,699
8
votes
1
answer
572
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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
2
answers
2k
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Tweetable way to see that Willmore energy is Möbius invariant?
Consider a compact orientable Riemannian manifold $M$ (without boundary) isometrically immersed into $\mathbb{R}^3$. The Willmore energy of $M$ is the functional
$$\mathcal{W} = \int_M H^2 dA$$
...
- 3,059
7
votes
1
answer
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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 ...
- 2,513
7
votes
3
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340
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Hyperbolic space embeds into Wasserstein space
Fix a positive integer $n$, let $\mathbb{H}^n$ be the $n$-dimensional hyperbolic space, $r>0$, $x\in \mathbb{H}^n$ and consider the closed (compact) geodesic ball $B_{\mathbb{H}^n}(x,r)$. Are ...
- 195
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1
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Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
- 356
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votes
2
answers
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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?
...
7
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2
answers
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Uniformizations of the bordered/punctured Riemann surfaces
The uniformization theorems of Riemann surfaces state that any Riemann surface can be constructed by an action of some group on some space. It is quite hard to find materials relating different ...
- 989
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2
answers
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injectivity radius of hyperbolic surface
Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?
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