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24 votes
2 answers
1k views

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 ...
Neil Strickland's user avatar
24 votes
3 answers
1k views

Hyperbolic Coxeter polytopes and Del-Pezzo surfaces

Added. In the following link there is a proof of the observation made in this question: http://dl.dropbox.com/u/5546138/DelpezzoCoxeter.pdf I would like to find a reference for a beautiful ...
Dmitri Panov's user avatar
  • 28.9k
19 votes
1 answer
902 views

Locus of equal area hyperbolic triangles

Henry Segerman and I recently considered the following question: Given a fixed area $A < \pi$ and two fixed points in the upper half-plane model for hyperbolic $2$-space, what is the locus of ...
Grant Lakeland's user avatar
18 votes
2 answers
4k views

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 ...
Boyu Zhang's user avatar
18 votes
2 answers
1k views

Hyperbolic Volume and Chern-Simons

In the paper ``Analytic Continuation Of Chern-Simons Theory'' (arXiv:1001.2933) Witten postulates that hyperbolic volume of 3-dimensional manifold coincides with the value of the Chern-Simons ...
d1-d5's user avatar
  • 183
13 votes
3 answers
2k views

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-...
user82102's user avatar
  • 133
12 votes
4 answers
991 views

Reference for the proof that Möbius transformations extend to isometries of hyperbolic 3-space

Consider the group $\operatorname{PSL}(2,\mathbb C)$ acting by Möbius transformations of the Riemann sphere. It is known that this action can be extended to an action on the unit ball which is ...
Ilya Gekhtman's user avatar
11 votes
3 answers
823 views

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 $M_1\# M_2$ ...
Marc Kegel's user avatar
  • 1,314
11 votes
4 answers
649 views

Introductory textbook on geometry of hyperbolic space

I am looking for an introductory textbook to the geometry of the hyperbolic space $\mathbb{H}^n$. The book should include explicit description of geodesics and horospheres in various models (...
10 votes
4 answers
667 views

Reference for shortest educational path to (Riemannian) hyperbolic plane

I am teaching an undergraduate class for math majors on axiomatic geometry, culminating in the proof that hyperbolic geometry is equiconsistent* with Euclidean geometry. I would like to make an end-of-...
David Steinberg's user avatar
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. ...
Joseph O'Rourke's user avatar
9 votes
6 answers
4k views

Books for hyperbolic geometry ( surfaces ) with exercises?

what are good books on hyperbolic geometry/hyperbolic surfaces that have good number of exercises, just to get a good understanding of the literature . I know John Ratcliffe's book will be one of them,...
Analysis Now's user avatar
  • 1,471
9 votes
1 answer
480 views

What is the complex structure on the boundary torus of a hyperbolic knot complement?

Let $K$ be a hyperbolic knot in $\mathbb S^3$. Restrict the corresponding representation $\pi_1(\mathbb S^3\setminus K)\to\operatorname{PSL}(2,\mathbb C)$ to the fundamental group of the boundary (...
John Pardon's user avatar
  • 18.7k
9 votes
0 answers
331 views

Connections between spectral geometry and critical point/Morse theory

I am researching electrostatic knot theory, which is essentially the theory of harmonic functions on knot complements. I want to understand the number of critical points of the electric potential, ...
maxematician's user avatar
9 votes
0 answers
360 views

Phillips-Sarnak conjecture in higher dimension

The Phillips-Sarnak conjecture states that for a generic Fuchsian lattice the space of Maass cusp forms is finite-dimensional. Generic here means in particular non-uniform, non-arithmetic, no special ...
Maik Köster's user avatar
8 votes
2 answers
631 views

Teichmüller space on non-orientable closed surfaces

It is known that any closed orientable surface of genus $g \geq 2$ admits a hyperbolic metric, and the Teichmüller space of such metrics has dimension $6g - 6$. I was wondering if there is a ...
Curious's user avatar
  • 81
7 votes
1 answer
957 views

Relations between some works by Deligne-Mostow and Thurston

Happy new year 2016! A coworker and I are interested in the relations between the works of Deligne and Mostow ([DM] and [M]) on the monodromy of Appell-Lauricella hypergeometric functions (Publ. ...
Elbabak's user avatar
  • 347
7 votes
3 answers
413 views

Best source for classification of right-angled hyperbolic hexagons

A standard fact that underlies the Fenchel-Nielsen coordinates on Teichmuller space is the fact that for all triples $(a,b,c)$ of positive real numbers, there exists a unique hyperbolic hexagon whose ...
Lisa's user avatar
  • 71
7 votes
1 answer
343 views

For a 3D Apollonian packing, do we really know that the Hausdorff dimension of the complement is approximated by the growth rate of curvature?

The fractal dimension of the 3D Apollonian packing is computed in this paper. In the introduction, the authors cite three of Boyd's paper (Ref 2, 5, 6) to support that the fractal dimension (...
Hao Chen's user avatar
  • 2,581
7 votes
0 answers
1k views

Closed geodesics on a closed, negatively curved Riemannian manifold

I have been searching for a while for a proof of the following fact: For a closed Riemannian manifold, all of whose sectional curvatures are negative, each free homotopy class of loops contains a ...
Clark's user avatar
  • 71
6 votes
1 answer
800 views

Geometrization & JSJ decomposition with boundary

Is there any paper where I can find a good explanation of the JSJ decomposition, the geometrization theorem and the relations between them when the manifold has nonempty (and non necessarily toroidal) ...
Antonio's user avatar
  • 125
6 votes
1 answer
414 views

Uniformization of a plane minus cantor set

Let $\mathbb{D}$ be the unit disk endowed with the Poincaré metric and $G$ be a Fuchsian group such that the hyperbolic surface $\mathbb{D}/G$ is homeomorphic to the plane minus a Cantor set. ...
Pablo Lessa's user avatar
  • 4,304
6 votes
1 answer
157 views

Configurations of $n$ points modulo isometries of the ambient space

Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate ...
bjw's user avatar
  • 63
6 votes
1 answer
338 views

Reference request: embedding the hyperbolic triangulation in $\mathbb{R}^3$

Let $T_d$ be the infinite valence $d$ triangulation of the hyperbolic plane, where each triangle is equilateral and $d \ge 7$. Question: Is there an isometric embedding from $T_d \to \mathbb{R}^3$? ...
Geoffrey Irving's user avatar
6 votes
1 answer
434 views

Previous work on this generalization of continued fractions?

The 2x2 matrix representation of a continued fraction makes it clear that we're multiplying together a bunch of group elements. Inversion is essentially freely adjoining a generator to a Coxeter ...
Mike Stay's user avatar
  • 1,532
6 votes
0 answers
200 views

Spectral theory for Dirac Laplacian on a funnel

I would like to study the spectral theory of the Dirac Laplacian on a non-compact quotient of the hyperbolic plane by a discrete group (I am particularly interested in the simple case where the ...
harlekin's user avatar
  • 313
5 votes
3 answers
432 views

What is the limit set of a hyperbolic lattice?

My claim is as follows: Let $\Gamma$ be a discrete subgroup of $\operatorname{Isom}(\Bbb{H}^{n})$, the isometries of hyperbolic $n$-space. If $\Gamma$ is a lattice in $\operatorname{Isom}(\Bbb{H}^n)...
Joe Wells's user avatar
  • 195
5 votes
3 answers
295 views

Teichmuller space for surface with cone points

Working on my current research problem, Teichmuller spaces for surfaces with cone points have come into play. It's fairly easy to formulate some of the definitions, a few basic results, etc. To be ...
user470881's user avatar
5 votes
3 answers
593 views

Who first used the cross-ratio to describe shapes in hyperbolic geometry?

I was reading this Wikipedia article today:https://en.wikipedia.org/wiki/Shape#Similarity_classes and I realized that it strongly resembles the use of coss-ratios as "shape parameters" in hyperbolic ...
Brian Rushton's user avatar
5 votes
1 answer
439 views

Rotation part of short geodesics in hyperbolic mapping tori

Otal [Sur le nouage des géodésiques dans les variétés hyperboliques. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 7, 847--852.] showed that "short" simple closed geodesics in 3-dimensional ...
b b's user avatar
  • 1,601
5 votes
1 answer
394 views

closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface

Could you please recommend me some references for proofs of this fact: "closed geodesics are dense in both hyperbolic surface and unit tangent bundle of hyperbolic surface". Thanks in advance!
Markiff's user avatar
  • 333
5 votes
1 answer
142 views

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 ...
aglearner's user avatar
  • 14.3k
5 votes
1 answer
241 views

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:...
Neil Strickland's user avatar
5 votes
1 answer
291 views

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 $3$...
user126154's user avatar
5 votes
1 answer
353 views

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} &...
Lasse Rempe's user avatar
  • 6,548
4 votes
2 answers
479 views

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 ...
matgaio's user avatar
  • 345
4 votes
2 answers
998 views

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 ...
Alex 'qubeat''s user avatar
4 votes
1 answer
198 views

Examples of hyperbolic manifolds of dimension $\geq$ 3 with disjoint totally geodesic hypersurfaces

I am hoping to find examples of compact hyperbolic manifolds with at least 2 disjoint totally geodesic hypersurfaces. Ideally, I would like examples in dimension at least 4, though 3-dimensional ...
ಠ_ಠ's user avatar
  • 6,025
4 votes
1 answer
307 views

The Weyl law for lengths

For what I know, this must be a standard fact, but I can't spot it in the literature I have on hands. What is the asymptotic of the geodesic lengths spectrum for the modular surface $X(1)$? (That is, ...
Alex Gavrilov's user avatar
4 votes
2 answers
378 views

Comparing two Delaunay tessellations on a hyperbolic surface

Let $S$ be a closed hyperbolic surface (i.e. a compact Riemann surface of genus $\geq 2$) and let $P=\{p_1,\ldots,p_m\}$ be a non-empty finite subset of $m$ points in $S$. Let $\pi:\mathbb H\...
Lucien's user avatar
  • 838
4 votes
1 answer
283 views

Mapping the hyperbolic plane onto the interior of a disk

In Chapter II of his book Non-Euclidean Geometry (1961; first published in Polish, 1956), Stefan Kulczycki defines a mapping of the hyperbolic plane onto the interior of a disk. Its construction ...
Firestone's user avatar
4 votes
0 answers
433 views

Convex core and geometric finiteness of negatively curved manifolds

I am reading a paper on hyperbolic geometry where they are using the concept of "convex core" in the context of "geometric finiteness". Roughly, this means (from Definition F4 of a ...
user481559's user avatar
4 votes
0 answers
88 views

What is the explicit relationship between the shape parameters and the holonomy of a hyperbolic ideal triangulation?

Let $K$ be a hyperbolic knot in $S^3$. One way to describe the hyperbolic structure is to give a discrete, faithful representation $\pi_1(S^3 \setminus K) \to \operatorname{PSL}_2(\mathbb C)$, the ...
Calvin McPhail-Snyder's user avatar
4 votes
0 answers
670 views

Thales' Theorem for Hyperbolic Geometry [duplicate]

In Euclidean geometry Thales' Theorem says that if you view a diameter of a circle from any point on the perimeter it occupies exactly $90$ degrees in your field of view. More generally for any ...
Pablo Lessa's user avatar
  • 4,304
3 votes
2 answers
950 views

hyperbolic 3-manifold of finite volume

Is there a complete description of hyperbolic 3-manifold of finite volume ? Or similarly a classification of finitely generated torsion free subgroups of $PSL(2,\mathbf{C})$ with finite covolume? ...
mathphys's user avatar
  • 1,629
3 votes
3 answers
1k views

Books that discuss spectral graph theory and its connection to eigenvalue problems in hyperbolic geometry

Hello, Could you name a couple of books or downloadable lecture notes that discuss spectral graph theory and its connection to spectral problems in hyperbolic Riemann surfaces ? You could also ...
Analysis Now's user avatar
  • 1,471
3 votes
1 answer
142 views

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 ...
asv's user avatar
  • 21.8k
3 votes
2 answers
420 views

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 (...
user2022's user avatar
3 votes
1 answer
158 views

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 ...
Xin Nie's user avatar
  • 1,804
3 votes
1 answer
375 views

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.
KAK's user avatar
  • 613