Questions tagged [hyperbolic-geometry]
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863
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What does the trace of a loxodromic Mobius transformation tell us about how it rotates?
A matrix $X=\begin{pmatrix}a&b\\c&d\end{pmatrix}\in\mathrm{PSL}_2(\mathbb{C})$
acts isometrically on the upper half-space model $\mathbb{H}^3$
via isometric extension of the Mobius ...
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3
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In the $\mathbb{H}^3$ upper half space model, is a hemiellipsoid perpendicular to the plane at infinity a minimal surface?
Given a Jordan curve on the $\mathbb{H}^3$ boundary at infinity, there is a minimal surface (topological disk) in $\mathbb{H}^3$ with the curve as its asymptotic boundary (page.mi.fu-berlin.de/...
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Blaschke Condition for hyperbolic lattices
For $r$, $s$, small positive integers, do the complex numbers on the unit disc (without the hyperbolic metric) corresponding to the vertices of the hyperbolic tiling with Schläfli symbol $\{r,s\}$ ...
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2
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678
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Volume form on a hyperbolic manifold with geodesic boundary
Let $M$ be a compact connected orientable Riemannian $n$-manifold with boundary $\partial M\ne\emptyset$. Since $H^n(M,\mathbb R)=0$, the connecting morphism $\delta: H^{n-1}(\partial M,\mathbb R)\to ...
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228
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Hyperbolic Space, Lattice Isometries, and Polyhedral Fundamental Domains
Let $L$ be a lattice of signature $(1,n)$. Suppose I have a (probably infinite index) subgroup $\Gamma\subset O^+(L)$ of the isometries of $L$ which preserve the positive cone $\mathcal{C}^+\subset L\...
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2
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438
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Subgroups of hyperbolic groups
Let $G$ be a finitely generated hyperbolic group, and let $H \leq G$ be a subgroup whose profinite completion is finitely generated. Must $H$ be finitely generated?
In view of Ian Agol's answer, I ...
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Compact open topology on the space of geodesics
I'm new in the field, so I'm sorry in advance if my question is too naive.
Let's consider $S$ a surface of genus $g\ge 2$ with an hyperbolic metric $g$. Let's call $\mathcal{S}(S)$ the set of closed ...
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1
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Lamination as limit of arcs
I am reading Bonahon's notes on closed curves, in particular the part about hyperbolic laminations. In his notes Bonahon illustrates some examples as why laminations should be "limit curves" on ...
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How bad is the modular space?
I'm wondering if there is some results about the quotient space $\mathbb{H}^{3}/PSL(2,\mathcal{O}_{K})$?
Do we know something about its homology or homotopy groups ?
$\mathbb{H}^{3}$ is the hyperbolic ...
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Intuition for Zagier's theorem for $\zeta_K(2)$
In 1986, Don Zagier generalized Euler's theorem ($\zeta_\mathbb{Q}(2)=\pi ^2 /6$) to an arbitrary number field $K$:
$$\zeta_K(2)=\frac{\pi^{2r+2s}}{\sqrt{|D|}}\times \sum_v c_v A(x_{v,1})...A(x_{v,s})...
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2
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Why simple closed curves are dense in $\mathcal{PML}_0(S)$?
I have another question about laminations on surfaces. As usual let $\mathcal{S}$ be the set of homotopy classes of simple closed curves in $S$ and $\mathcal{PML}_0(S)$ be the set of projective ...
user47069
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Why is $\mathcal{PML}_0(S)$ compact?
I'm starting to study geodesic laminations on hyperbolic surfaces and in particular I'm focusing my attention on $\mathcal{PML}_0(S)$, the space of projective classes of measured geodesic laminations ...
user47069
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The hyperbolic metric on a flat surface
Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ($k\in\mathbb{N}$)...
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3
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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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Is there a formula for the A-model partition function in terms of hyperbolic structure?
The A-model partition function is a topological invariant of any hyperbolic surface. So, what is this invariant?
I'm not sure if there is a way to normalize these since the usual perscription $Z(S^2) ...
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Algorithm to generate hyperbolic metric on a compact surface
Let $F$ be a compact surface of genus $g$ with generators $a_1,b_1,\ldots ,a_g,b_g$ with relation $[a_1,b_1]\cdots [a_g,b_g]=1$. We can also consider surfaces with boundary in which case the ...
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Length and laplacian spectrum for quasi-fuchsian manifold
It is well known that, in the case of finite area hyperbolic surfaces, the length sprectrum (the collection of length of all closed geodesics) and the spectrum of the laplacian (acting on functions) ...
user50806
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A group acting acylindrically on a fine hyperbolic graph with infinite edge stabilizers
I am looking for an example of a group $G$ that acts (cocompactly and) acylindrically on a hyperbolic graph $\Gamma$, such that
a) the graph $\Gamma$ is fine,
b) $\Gamma$ is not a tree,
c) not all ...
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Is the heat kernel for the hyperbolic plane uniformly continuous in $t\in(0,\infty)$?
let $\mathbb{H}$ be the hyperbolic plane and let $k(t,x,y)$ be the associated heat kernel.
I am wondering, if for any fixed $y\in M$ and $\epsilon >0$ the function $u_t(x):=k(t,x,y)$ is continuous ...
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Conditions on the hierarchy for Thurston's hyperbolization theorem
From my understanding the proof of Thurston's hyperbolization theorem for Haken $3$--manifolds consists of cutting the manifold along a hierarchy (collection of incompressible, $\partial$-...
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Heegard genus of hyperbolic Haken 3-manifolds
Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?
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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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SnapPea for the uninitiated
SnapPea (http://www.math.uic.edu/~t3m/SnapPy/) is a program with extensive facilities for doing various kinds of calculations with hyperbolic 3-manifolds. The official documentation assumes that the ...
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What is the metric on the Fuchsian model? [closed]
Let $\mathbb{H}$ be the upper half plane, and $\Gamma < SL(2, \mathbb{R})$ be a Fuchsian group. How is the distance between any two points $x, y \in \mathbb{H} / \Gamma$ in the Fuchsian model ...
user76098
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Maximum relator and hyperbolicity
It is a well-known theorem that a finitely generated group is hyperbolic if and only if it admits a finite Dehn presentation. To prove the "only if" direction one proceeds roughly as follows:
Suppose ...
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local quasi geodesics in hyperbolic spaces
I asked this question on math stackexchange (see here) but didn't get any answer so I thought I post it here too.
We have the following two well-known Theorems:
T1) For all $\delta > 0, \lambda ...
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Is the length function associated with the twist parameter an increasing function?
Let $S$ be a closed hyperbolic surface and $x$ be an oriented simple closed curve in $S$. Let $y$ be an oriented closed curve such that the geometric intersection number between $x$ and $y$ is ...
- 1,703
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Diameter of hyperbolic 3-manifolds
Is there a good method to estimate the diameter of a closed hyperbolic 3-manifold?
I am particularly interested in know the diameter of the Weeks manifold.
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0
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Immersed surfaces in Hyperbolic 3-manifolds
Given a hyperbolic 3-Manifold $M=\Gamma_{0}\setminus\mathbb{H}^3$, and a smooth, connected, compact immersed negatively curved surface $\Sigma=\Gamma\setminus\widetilde\Sigma\subset M$, where $\...
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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 ...
- 1,337
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Complex structure and antipode map on the space of measured geodesic laminations
Fix a closed hyperbolic surface $S$, which represents a point in the Teichmüller space $\mathcal{T}$ of the underlying topological surface.
Thurston's earthquake theorem implies an identification ...
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1
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Infinitesimal deformations of fake projective planes (or ball quotients)
This question is related to the answer I gave to this MO question. What I'm asking is probably well-known to the experts in the field, and I apologize in advance if this turns out to be trivial.
By ...
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Hyperbolic knot complement groups and relative dimension
Let $G$ be a the fundamental group of a hyperbolic knot complement.
Then $G$ is hyperbolic relative to a subgroup $P\cong \mathbb Z \oplus \mathbb Z$.
The knot complement has a $2$-dimensional spine ...
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Hyperbolic manifold of dim 3 with finite volume.
The geometrization Theorem for 3-manifolds classifies all oriented compact (without boundary ) 3-manifolds. Is there a theorem which classifies all hyperbolic oriented 3-manifold (without boundary ) ...
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Decomposition of hyperbolic surfaces near cusps into annuli
Let $C=\mathbb{H}/\Gamma$ be a hyperbolic surface and $c$ a cusp of this sruface. In the paper "Billiards and Teichmüller curves on Hilbert modular surfaces" by C. McMullen, it is claimed that near ...
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Real slices of Minkowski space, using a complex quadratic form
Ordinary Minkowski space is $\mathbb{R}^{3,1}:=(\mathbb{R}^4,\phi)$
where $\phi:\mathbb{R}^4\rightarrow\mathbb{R}$
is a quadratic form of signature $(3,1)$.
Lying within this is a hyperboloid model ...
- 2,438
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2
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459
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Question about the Weeks Manifold
I am wondering about the existence of incompressible surfaces in the Weeks manifold. Is this space a Haken manifold?
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What sort of models did Bolyai and Lobachevsky use to demonstrate the consistency of their models of non-Euclidean Geometry?
As is well-known, in the 1820s both Bolyai and Lobachevsky showed, at long last, the independence of the Parallel Postulate from the rest of the axioms of Euclidean geometry by developing what we now ...
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Kleinian groups containing an isomorphic copy of itself
Is there any example of a Kleinian group (acting on $\mathbb{H}^n$, $n \ge 3$) that contains a finite index isomorphic copy of itself? Here I don't consider Kleinian groups that only have parabolic ...
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Decay of cusps in geometrically finite groups
Let $X=\mathbb{H}^{n}/\Gamma$ be a quotient of hyperbolic space of a geometric finite subgroup. Let $\mu$ be the Bowen-Margulis measure on the unit tangent bundle, and $m$ its projection to $X$.
Fix ...
- 2,490
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Does this count as a canonical decomposition for non-elementary hyperbolic 3-orbifolds?
Let $\Gamma$ be a Kleinian group and let $\mathbb{H}^3$ be the upper half-space model for hyperbolic 3-space.
Then $\mathbb{H}^3/\Gamma$ is an orientable hyperbolic 3-orbifold (with the group action ...
- 2,438
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2
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Negatively curved metrics minimizing the length of a homotopy class of simple closed curves
Good afternoon everyone !
I have the following question of Riemannian geometry :
Let $M$ be a smooth closed orientable manifold of dimension at least $3$, and let $\mathcal{T} = \{ $ smooth ...
- 2,636
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1
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Reduction of self-intersections without reducing the geometric intersection
Let $F$ be a hyperbolic surface. Given a closed curve $a$, let $\bar{a}$ denotes the free homotopy class of $a$. Let $i(\bar{a},\bar{b})$ denotes the geometric intersection number and $i(\bar{a})$ ...
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Harmonic spinors on closed hyperbolic manifolds
Does anyone know an example of a closed spin hyperbolic manifold of dimension 3 or greater such that the kernel of the Dirac operator is non-trivial?
I'm mainly interested in the 3-dimensional case ...
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Teichmuller distance between isospectral riemann surfaces
Let $S$ be a surface of negative Euler characteristic (for simplicity let's assume $S$ to be closed), and let $\mathcal{M}(S)$ denote the moduli space of hyperbolic surfaces homeomorphic to $S$.
...
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Is there a universal straightedge and compass construction of a segment incommensurable to a given one in the hyperbolic plane?
"Universal" means that the construction steps are independent of the length of the given segment. In the Euclidean plane one can take the diagonal of a square built on it. Without the "universal" the ...
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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 ...
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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?
- 2,490
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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 ...
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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 ...
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