Questions tagged [hyperbolic-geometry]
The hyperbolic-geometry tag has no usage guidance.
885 questions
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Busemann-Feller lemma in hyperbolic space
The classical Busemann-Feller lemma in Euclidean space says the following.
Let $K\subset \mathbb{R}^n$ be a closed convex set.
Then
for any point $x\in \mathbb{R}^n$ there exists unique nearest point ...
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Learning roadmap for Lorentzian geometry
I am asked the question in MSE, but did not get an answer. I hope that this question is appropriate for MOF.
I am interested in Hyperbolic Geometry and its significance in low dimensional geometry (...
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618
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Schwarz Lemma in terms of conformal surfaces or holomorphic curves?
Scharwz Lemma in its general form says that any holomorphic map between hyperbolic surfaces is contracting.
Noting that Riemann surfaces admit a unique metric of constant curvature -1, I wonder if we ...
user2529
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Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures
In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to ...
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Reference request: geometric finiteness of Fuchsian groups
My limited knowledge on hyperbolic geometry suggests me that the following proposition should be true (please correct me if I'm wrong):
Proposition. The convex core of a complete hyperbolic surface ...
- 1,804
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2
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What is the homeomorphism from $\Gamma \backslash T_1 \mathbb{H}$ to $T_1(\Gamma \backslash \mathbb{H})$
Let $\mathbb{H}$ be hyperbolic plane, $\Gamma$ is a discrete subgroup of $PSL_2(\mathbb{R}$) so that $\Gamma \backslash \mathbb{H}$ is a compact hyperbolic surface. Maybe it will be very simple to you ...
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Intersection of closed geodesics in hyperbolic surface
This question may be easy but I could not come up with a proof.
Let $F$ be a hyperbolic surface of finite type (with finitely many boundary and finitely many puncture). Let $\gamma$ be a closed non-...
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Geodesic convexity of Dirichlet Fundamental Domains
My question is motivated by this question, and this answer to it. Below, let's consider the setup in that answer:
Let $M$ be a Riemannian manifold. Let $G\times M\to M$ be a proper action of a ...
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1
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Lengths of generators of surface group
Let $\Sigma$ be a closed genus $g\geq 2$ Riemann surface, which we equip with its unique constant curvature $-1$ hyperbolic metric. Let $\pi_1(\Sigma)$ be its fundamental group with respect to some ...
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Special kind of 3-manifolds
Is there an open connected orientable 3-manifold $M$ with the following properties:
$M$ admits a complete hyperbolic metric with finite hyperbolic volume.
$H_{i}(M,\mathbb{Z})=0$ for any $i>0$.
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δ-hyperbolic space
It is known that A δ-hyperbolic space is a geodesic metric space in which every geodesic triangle is δ-thin. (http://en.wikipedia.org/wiki/%CE%94-hyperbolic_space)
The question is that if we remove ...
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Guts of 3-manifolds for sutured manifolds and pared manifolds
I found the notion "guts of three-manifolds" unclear to me. There exists "sutured guts" and "pared guts" in the literature, the well definedness of both are vague to me.
...
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Geodesic laminations on the 4-punctured sphere
Let $S_{0,4}$ be the 4-punctured sphere equipped with a hyperbolic metric of finite volume (thus all punctures are cusps). Consider $\gamma$ to be a simple geodesic of $S_{0,4}$ (not necessarily ...
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Reference for triangle groups
Can anyone suggest to me some references for studying triangle groups? Especially the existence of finite index subgroups, subgroups isomorphic to fundamental groups of compact surfaces etc.
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Parabolic elements and hyperbolic elements in SL(2,R)
Let $\Gamma \subset \mathrm{SL}(2,\mathbb{R})$ be a lattice. If $N_1, N_2$ are a pair of independent parabolic subgroups contained in $\Gamma$, why must $\Gamma$ contain a hyperbolic element? By ...
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Volume of hyperbolic 3-manifolds with toroidal boundary
A hyperbolic 3-manifold has finite volume if and only if it is either closed or has toroidal boundary and it is not homeomorphic to $T^2\times I$.
This statement is from 3-Manifold Groups, page 18 (...
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Cusps of hyperbolic surfaces under finite covers
The following statement seems true, but I don't know a proof or a reference for it (and I would like one).
Let $\Gamma< \operatorname{PSL}(2,\mathbb R)$ be a nonuniform lattice with one cusp. We ...
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What is a half cusp in hyperbolic geometry?
I already asked this question on math.stackexchange, but it was suggested that I post it here as well.
The paper Devadoss, Heath, and Vipismakul - Deformations of bordered Riemann surfaces and ...
- 1,435
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Conformal invariants of planar pairs of pants
Consider a planar pair of pants
$$P = \left\lbrace z \in \mathbb{C}: |z| \le 1, |z-x| \ge r_1, |z+x| \ge r_2 \right\rbrace$$
where $-1 < -x-r_2 < -x+r_2 < 0 < x-r_1 < x+r_1 < 1$.
...
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In hyperbolic 3-orbifold with totally geodesic boundary case, is it true: rank(the fundamental group of boundary M)< or equal 2 rank(fundmental group of M)?
For a orientable three manifold M with totally geodesic boundary, this inequality is true. Because the rank of (fundemantal group of boundary M)=rank (homology group of boundary M )
then we use the "...
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Group action of $\text{SL}(2, \mathbb{C})$
Let's denote upper-half space model of hyperbolic 3-space identified with quaternions as $$\text{H}^{+}_3 = \biggr\{z+wj\in \mathbb{C}\oplus \mathbb{R}^{+}j\biggr\}\tag{1}$$
Then a group action $\rho :...
user515818
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1
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A closed curve can be homotopic to remove all intersections with a filling $\Gamma$ if it has zero geometric intersection numbers with $\Gamma$
Let $\Sigma$ be a compact oriented connected bordered surface other than the pair of pants. Let $\Gamma:=\{\gamma_i\}$ be a finite collection of simple closed curves on $\Sigma$ such that each ...
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Counterexample to mostow rigidity theorem
I am looking for an example of $M$ and $N$ two orientable hyperbolic complete without boundary 3-manifolds ( with infinite hyperbolic volume) such that $\pi_{1}M\cong \pi_{1}N$ but $M$ is not ...
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Coordinate transformation for the hyperbolic plane to the pseudo sphere
There are three common ways to represent the hyperbolic plane. One usually starts with the hyperboloid $x_1^2+x_2^2-x_3^2=-1$ embedded in a Minkowski space with metric $ds^2 = dx_1^2+dx_2^2-dx_3^2$. ...
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Is the development map in Hyperbolic geometry related to development in Cartan geometry?
I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ ...
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Classifying links with essential annuli in the complement as torus links
I've seen it claimed in several places, though never with a detailed proof, that every non-split link is either a hyperbolic, satellite, or torus link (see for example pg. 95 of Cromwell's "Knots and ...
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Kobayashi distance function on the upper half-space
I asked this question already in mathstackexchange but got no answer, so I ask it again here.
Let $\mathbb{H}_{g}$ be the Siegel upper half space, i.e., the set of complex symmetric $g\times g$ ...
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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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simplex in hyperbolic space and quadrics in projective space
I am trying to understand Goncharov's paper "Volumes of hyperbolic manifolds and mixed Tate motives", http://www.ams.org/journals/jams/1999-12-02/S0894-0347-99-00293-3/S0894-0347-99-00293-3.pdf
In ...
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Density of ends of long words in a hyperbolic group
Let $G$ be a Gromov hyperbolic group with a generating set $S$. For each $g \in G$, let $\xi_g$ be the point in the ideal boundary corresponding to the sequence $(g^n)$. Let $l_S(g)$ be the word ...
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Arithmetic Fuchsian lattices that are not finite index subgroups of Eichler orders?
Lindenstrauss' proof of AQUE (arithmetic quantum unique ergodicity) assumes that the Fuchsian lattice is an Eichler order or, if I understand it correctly, a finite index subgroup of an Eichler order. ...
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Euclidean surfaces with conical singularities and cusped hyperbolic surfaces
Let $S$ be a compact orientable surface endowed with a singular euclidean metric $g$, with $n$ conical singularities $x_1,\ldots,x_n$.
Construction 1: it is well-known that the conformal class $[g]$ ...
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Geometry and topology of Fuchsian character varieties
Consider the hyperbolic space, $\mathbb H^2$. A Fuchsian group is a discrete subgroup of $\text{PSL}(2,\mathbb R)$. We can generate tessellations, especially $\{p,q\} \;\text{tesellations}$ of $\...
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Comparing the areas of polygons via equidecomposability in the hyperbolic plane
It is well-known that in the Euclidean plane two simple polygons have the same area if and only if they are equidecomposable, i.e., can be decomposed into congruent triangles.
Question. Is an ...
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Does the Weeks manifold have the smallest volume among all finite volume oriented hyperbolic 3-manifolds?
It is a result of Chinburg-Friedman-Jones-Reid that the arithmetic hyperbolic 3-manifold of smallest volume is the Weeks manifold.
There is also a result of Milley that says that if $N$ is a closed ...
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On well separated circular regions in the Riemann sphere and complex polynomials
It started with a conjecture I had, see A statement on complex polynomials, which was false for $n \geq 3$, as shown by Noam D. Elkies in his answer there. The present post is an attempt to salvage ...
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Reference request for statement concerning free subgroups of $ \mathrm{SL}_2(\mathbb{Z}). $
I am interested in finding a reference for the following claim:
There exists a free subgroup $F_2$ of $\mathrm{SL}_2(\mathbb{Z})$ on two generators that does not contain any nontrivial unipotent ...
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Maximally symmetric hyperbolic 3-manifolds with finite volume
In physics, standard cosmology is build with simple maximally symmetric 3-manifolds (spacelike time-slices of constant curvature, e.g. $S^3$ or less popular the hyperbolic space $H^3$). Since $S^3$ ...
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Intersecting geodesics on a surface from non-intersecting geodesics
Let $a$ and $b$ be non-intersecting closed geodesics on a hyperbolic surface. Can these curves be homotopied to transversely intersect but still be geodesics?
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Question on a proof of density of periodic orbits
In page 215 and 216 of the book "Introduction to the Modern Theory of Dynamical Systems" by Anatole Katok, Boris Hasselblatt, there is a theorem stated as following:
Theorem: Let $\Gamma$ be a ...
user127549
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779
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English translation of M.-F. Vigneras "Arithmétique des algèbres de quaternions"
I am looking to understand a citation about the connection of quaternion algebra over number fields which when embedded into $\mathbb{C}$, leads to a discrete subgroup of $SL_2(\mathbb{C})$ which ...
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volume entropy and Hausdoff dimension
In relation to this question: Relation between volume entropy and Hausdorff dim of limit set?
Given a metric space $Z$ and a hyperbolic approximation $X := hyp_{r_0}(Z)$ (as defined for example here)....
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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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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
3
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1
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375
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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 ...
- 10.4k
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Parallel transport on a Hadamard manifold
Suppose, $X$ is a Hadamard manifold, i.e., a simply connected manifold of non-positive sectional curvature. Fix a point $w$ in $X$. Consider any three points $x, y, z$ in $X$. Let $\tau_{x, w}$ and $\...
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A question about hyperbolic double torus
Hi
I have a question about hyperbolic 2-torus, from now on donoted by $\Sigma_{2}$
Actually I've tried to prove that for a group $\Gamma \subset \textrm{Isom}^{+}(\mathbb{H}^{2})$ represented by $\...
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The version of Montel's theorem used in the proof of Jenkins-Strebel differential
Hello,
I am afraid that my main question might be a bit too elementary, but still I ask :
In short, my question is "what is the version of Montel's theorem for a family of holomorphic maps from an ...
- 1,471
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1
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187
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2-orbifolds that I expect to be hyperbolic, but they're nonnegatively curved
I'm considering some complex 1-dimensional/real 2-dimensional orbifolds that I expect to be hyperbolic. However, some of them seem to be Euclidean or spherical. Any thoughts what's going on here? Here ...
- 1,277
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What is the name of this geometric structure, where we identify each sphere of vision with the sphere at infinity?
If you consider hyperbolic $n$-space $H^n$, modeled by the open unit ball $B^n \subset \mathbb{R}^n$, then given any two distinct points $x_1$, $x_2$ in $H^n$, there is a natural way of identifying ...
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