Questions tagged [hyperbolic-geometry]
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885 questions
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Conformal boundary and cusp of figure-8 complement
As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete ...
- 309
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Short basis in $\pi_1$ on a hyperbolic surface of bounded diameter
First, some terminology. Let $(S,x)$ be a compact surface of genus $g>0$. A standard collection of loops $\gamma_1,\ldots, \gamma_{2g}$ based at $x$ is a collection of loops that cuts $S$ into a ...
- 14.3k
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Geodesics (with the same limit point) in a surface group of genus two
Consider a discrete Gromov-hyperbolic group $\Gamma$ (and its Cayley graph
$\mathcal{G}$ w.r.t. some generating set). The notion of
Gromov-boundary, indicated with $\partial\Gamma$,
is naturally ...
- 329
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Uniqueness of hyperbolic rescaling
Let $X$ be a compact oriented surface of genus at least two, equipped with a Riemannian metric $g$. By the uniformization theorem for Riemann surfaces, there is a conformal universal covering map $p:...
- 56.9k
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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 ...
- 1,713
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Bisectors in symmetric spaces
In William Goldman's book Complex Hyperbolic Geometry, bisector hypersurfaces play an important role. Given two points $x,y$, the bisector is the set of points equidistant from $x$ and from $y$. Do ...
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560
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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 ...
- 151
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Examples of compact hyperbolic surfaces/manifolds with very small or very large eigenvalues
Hello,
Is there any general ways to construct compact hyperbolic 2-manifolds with very small or very large eigenvalues ? Also, as a special case, can we construct a sequence of compact hyperbolic 2-...
- 1,471
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Why is the Teichmüller space of a surface homeomorphic to a component of the $\mathrm{PSL} (2, \mathbb R)$ character variety of its fundamental group?
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\PSL{PSL}$
I have a reference request for a proof for the following statement in the title:
The Teichmüller space $T_g$ of the surface $S_g$ of genus ...
- 153
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242
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Cancellation of elements in the Gromov boundary of a free group
Let $A$ be a finite set of free generators and their inverses and $F$ the free group generated by elements in $A$ (some call $A$ the alphabet of $F$). For each $g\in F$, use $\vert\,g\,\vert$ to ...
- 307
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Question about and good reference for Kahn and Markovic result
As far as I understand, the celebrated result of Kahn and Markovic about quasi-Fuchsian immersions of surfaces in hyperbolic 3-manifolds has the following corollary:
Let $M$ be a compact hyperbolic $3$...
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333
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Proof of homotopic essential simple close curves are isotopic
In the book by Benson Farb and Dan Margalit A primer on mapping class groups, Princeton Mathematical Series 49. Princeton, NJ: Princeton University Press (ISBN 978-0-691-14794-9/hbk; 978-1-400-83904-9/...
- 119
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Fenchel-Nielsen length-length coordinates on Teichmueller space?
Let $S$ be closed hyperbolic surface with genus $g\geq 2$. Let $Teich(S)$ be the Teichmueller space of $S$. It's well known that $Teich(S)$ is diffeomorphic to a (6g-6)-dimensional cell, where a ...
- 2,274
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Hyperbolicity of twist knots
In Example 1.4 of his book Invariants of Homology 3-Spheres, Saveliev noted that twist knots $K_n$ of type $(2n+2)_1$ are all hyperbolic. Here, $K_1$ is the figure-eight knot $4_1$ and $K_2$ is the ...
user160180
5
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248
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Explicit check of the invariance of the Weil-Petersson form
Using Fenchel-Nielsen coordinates, the Weil-Petersson metric can be written as
$\omega_{WP} = \sum_{i} d\ell_i \wedge d \tau_i,$
where $i$ is an index labelling the curves of a pants decomposition of ...
- 1,435
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342
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Relations between boundaries of groups acting on hyperbolic spaces with WPD elements
Let $(X,d)$ be Gromov-hyperbolic space and let $\Gamma$ be a finitely generated group acting on $\Gamma$ by isometries. Recall the following two definitions.
Say that the action is acylindrical if ...
- 2,090
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464
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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 $B$-...
- 15.4k
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Mapping-Class Groups of Subsurfaces of a Hyperbolic Surface
If $\mathcal{R}'$ is a closed subsurface of a hyperbolic surface $\mathcal{R}$, then there is an inclusion homomorphism between the mapping class groups:
$$\text{Mod}(\mathcal{R}')\longrightarrow \...
- 989
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Quantifying the monotonicity property of the hyperbolic metric
Suppose that $X$ and $Y$ are connected hyperbolic Riemann surfaces, with $X\subsetneq Y$. Let $\rho_{X,Y}$ be the density of the hyperbolic metric of $X$ with respect to that of $Y$; then $\rho_{X,Y} &...
- 6,548
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Monopole classes on hyperbolic 3-manifolds
Let $M$ be a closed hyperbolic $3$-manifold, and $e \in H^2(M)$ an integral cohomology class which is the first Chern class of a $Spin^c$ structure on $M$. Suppose there is a solution to the monopole ...
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Intersection of orbits of earthquake flow on Teichmüller space
Let $\Sigma$ be a closed oriented surface of genus $g\geq2$. We consider $\mu$ and $\nu$ two filling measured laminations on $\Sigma$. (We say that $\mu$ and $\nu$ fill $\Sigma$ if $\Sigma\setminus(\...
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173
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Counterexamples to the Ahlfors measure conjecture in higher dimensions
Let $\Gamma<SO(3,1)$ be a finitely generated, discrete group of isometries of $\mathbb H^3$. By work of Agol, Calegari, Canary, and Gabai, the limit set of $\Gamma$ is either the entire sphere $S^2\...
- 327
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Are there examples of hyperbolic manifolds with finite Bowen-Margulis measure and fundamental group which is not relatively hyperbolic?
It is well known that a geometrically finite hyperbolic manifold (quotient of $H^n$) has finite Bowen-Margulis measure.
Marc Peigné [1] constructed examples of geometrically infinite hyperbolic ...
- 481
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183
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Finitely generated nilpotent groups as cusp groups
I recently learned about the following question, asked by I. Kapovich :
Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
- 2,090
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691
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Visual boundary vs. ideal boundary of hyperbolic manifolds?
I apologize in advance if this is elementary; I have very little experience with hyperbolic manifolds, but I'm using hyperbolic manifolds for part of a current project.
Given a discrete torsion-free ...
- 6,025
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532
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How can we explicitly verify a canonical Dirichlet domain for this hyperbolic punctured torus
This is a follow-up question to a question from math.stackexchange: https://math.stackexchange.com/q/1436253/67563
Had it not been for the exchange there between myself and @Lee_Mosher in the comments ...
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Barycentric interpolation in hyperbolic triangles
Let $T$ and $T'$ be triangles in the hyperbolic plane $\mathbb{H}^2$, denote by $A, B, C$ and$A', B', C'$ their vertices respectively. Let $f : T \to T'$ be the unique "barycentric interpolation" that ...
- 1,347
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Reference request: 3-dimensional Mobius transforms
I am working on a project that I suspect requires calculations involving Möbius transformations on 3-dimensional space $\mathbb{R}^3$, identified with the quaternions $\mathbb{H}$ with $k$-component ...
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Fundamental groups of closed hyperbolic 3-manifolds are freely indecomposable
I believe the following statement is true, and I've even seen it referenced here. Could someone point me to a proof?
The fundamental group of a closed hyperbolic 3-manifold is not a free product.
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Hyperbolic right-angled hexagon
Is there any equation looks like Cagnolli’s first formula(for hyperbolic triangle ABC):
$$\sin\frac{S}{2}=\frac{\sinh\frac{a}{2}\sinh\frac{b}{2}\sin C}{\cosh\frac{c}{2}}$$
for hyperbolic right-angled ...
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Tiling of genus 2 surface by 8 pentagons
In theses these notes, Example 5.6, it is said that there is a "symmetric tiling of a genus 2 surface by 8 right-angled hyperbolic pentagons".
Question 1: What does this tiling look like?
Question 2:...
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Dilogarithm, tetrahedrons, and hyperbolic space
The Bloch-Wigner function $D(z)$ gives the volume of an ideal tetrahedron in the hyperbolic space $\mathbb{H}^3$. Here $z$ is the cross-ratio $(z_1,z_2,z_3,z_4)$ parametrizing the tetrahedron in $\...
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439
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Simple Closed Hyperbolic Geodesics on Punctured Spheres
Thinking of $\mathbb {CP^1}$ as the sphere $S^2\subset\mathbb R^3$, we can define the notion of a circle on it to be a subset that is got by a hyperplane section of $S^2$ inside $\mathbb R^3$. This ...
- 1,761
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Knot complements with respect to Thurston's 8 geometries
We can put knot complements in three buckets: hyperbolic knots, satellite knots, and torus knots.
Can this classification be made with looking at the (complete) metric on the complement, if it has ...
- 1,465
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3
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teichmuller geodesics and hyperbolic mapping torus
Given a pseudoanosov map $\phi$ of a surface $S$, there is a geodesic $\sigma$ in Teichmuller space (with the teichmuller metric) that is an axis for $\phi$, In other words, $\phi$ acts as a ...
- 1,101
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Hyperbolic three-manifolds that fiber over the circle
Let $f$ be a pseudo-Anosov mapping class of a closed, connected, and oriented genus $g > 1$ surface. Let $M(f)$ be the corresponding hyperbolic three-dimensional mapping torus of $f$. Is the length ...
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Visibility spaces and Gromov hyperbolicity
I would like to ask the community for a reference on the following subject: is there some thing as an equivalence between the definitions of Uniform Visibility manifolds and Gromov $\delta$-hyperbolic ...
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Quasiconformal harmonic extension of a quasi-symmetric map on $S^1$
Hello ,we know that for given $h:S^1\to S^1$, we can solve the Dirichlet problem on $\bar{D} $ with the boundary value $h$ and in fact this extension, which is the complex harmonic extension $H=E(h) $ ...
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Fundamental group of a thick part of hyperbolic manifold
Let $M$ be a complete hyperbolic manifold of dimension $n$, let $\varepsilon=\varepsilon_n$ be the Margulis constant. Let $M_{[\varepsilon,\infty)}$ be the thick part of $M$ with respect to $\...
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When existence of loxodromic, WPD elements implies an action is acylindrical
Definitions
Say that $(X,d)$ is a $\delta$-hyperbolic space and that $G$ is a finitely generated group acting on $X$ by isometries.
Recall that an action of $G$ on $X$ is called acylindrical if the ...
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Mostow rigidity for complex hyperbolic manifolds
A Riemannian manifold $(X,g)$ is hyperbolic if the sectional curvatures are constant and negative. A theorem of Mostow says that these manifolds are determined by their fundamental group.
Theorem (...
- 1,231
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Dirichlet polyhedra for hyperbolic manifolds
Let $H$ be a simply-connected, complete
space of constant negative curvature, that
is, a hyperbolic space, $\Gamma$ a discrete group of
isometries, and and $M=H/\Gamma$ its quotient space;
we assume ...
- 9,237
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Two curves filling a surface
Let $S$ be a closed surface of genus $g \geq 2$. Do there exist two simple closed curves filling $S$?
Definitions:
Two closed curves $\alpha$, $\beta$ fill $S$ if they have minimal intersection and $...
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Relationship between hyperbolicity in group theory and hyperbolicity in geometry
Could somebody teach me about the relationship, if any, between hyperbolicity in groups (in Gromov sense) and hyperbolicity in 3-dimensional orbifolds? To be more specific, let Q be a 3-dimensional ...
4
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2
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998
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A construction of generators of discrete subgroups of SL(2,R)
I know about geometrical method of construction of discrete subgroups of $SL(2,\mathbb{R})$ using Lobachevsky plane (e.g. B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, Modern Geometry --- Methods and ...
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Visualizing hyperbolic metric of punctured sphere
Uniformization of the 3-punctured sphere generates a "pants" configuration with three legs narrowing down to cusps. This is supposed to have a metric of constant negative curvature, and I can see this ...
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hyperbolic orbifolds of small area
Is there a list of 2-dimensional hyperbolic orbifolds obtained from reflection groups (such as the double of a hyperbolic triangle with angles $\pi/p$, etc.) of small area, for instance area smaller ...
- 16.6k
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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 ...
- 1,601
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339
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Pseudoanosov mapping torus and length of curves.
Let $M_{\phi}$ be a hyperbolic mapping torus coming from a pseudo-Anosov map $\phi$ in a surface $S$. Is there any way to estimate the length of the geodesic representing a given curve in the surface ...
- 1,101
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995
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Connection 1-forms of a Riemannian metric and the norm of the Hessian and ( seemingly ) two different definitions of Hessian and its norm
In the paper "On Quasiconformal Harmonic Maps " (link here) by L. F. Tam and T.Y.H. Wan, Pacific Journal of Mathematics, vol 182, no 2, 1998, in section 1, they define the Hessian of a function $f :H^...
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