The hyperbolic-geometry tag has no usage guidance.

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**1**answer

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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, $\partial$-...

**7**

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**2**answers

212 views

### Heegard genus of hyperbolic Haken 3-manifolds

Is there an example of a closed Haken hyperbolic 3-manifold of Heegaard genus 2?

**5**

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**1**answer

256 views

### 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: http://math.stackexchange.com/q/1436253/67563
Had it not been for the exchange there between myself and @Lee_Mosher in the comments ...

**23**

votes

**2**answers

671 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 ...

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**0**answers

91 views

### 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 ...

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**0**answers

89 views

### 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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votes

**1**answer

140 views

### 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, \...

**2**

votes

**1**answer

44 views

### 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 ...

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**1**answer

223 views

### 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.

**1**

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**0**answers

80 views

### 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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**0**answers

86 views

### 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 ...

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55 views

### 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**answer

194 views

### 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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votes

**1**answer

203 views

### 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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**0**answers

141 views

### 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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**1**answer

153 views

### 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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**1**answer

91 views

### 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 ...

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**2**answers

287 views

### Question about the Weeks Manifold

I am wondering about the existence of incompressible surfaces in the Weeks manifold. Is this space a Haken manifold?

**16**

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**1**answer

397 views

### 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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**2**answers

288 views

### 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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**1**answer

151 views

### 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 ...

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**1**answer

114 views

### 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 ...

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**2**answers

150 views

### 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 ...

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**1**answer

140 views

### 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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**1**answer

844 views

### 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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**2**answers

180 views

### 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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**0**answers

112 views

### 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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**1**answer

172 views

### 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 ...

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votes

**2**answers

195 views

### 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?

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**2**answers

162 views

### 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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**2**answers

159 views

### 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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**0**answers

275 views

### When is a word metric on a CAT(-1) group a bounded distance from the orbit map of an isometric action on some CAT(-k) metric space?

Let $\Gamma$ be a group admitting a discrete and cocompact action on a CAT(-1) space.
Let $d$ a word metric on $\Gamma$ coming from some finite set of generators.
My question is:
Does there exist a ...

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votes

**2**answers

242 views

### Some general properties of arithmetic groups of simplest type

I'm working in the area of arithmetic Kleinian groups (as discrete groups of motions of hyperbolic 3-space). For the more general case of hyperbolic $n$-space, there is a particular class of ...

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votes

**1**answer

113 views

### Geometric realization of an abstract triangulation of the plane

Can every abstract simplicial complex whose geometric realization is homeomorphic to $\mathbb{R}^2$ be realized by a rectilinear triangulation of the Euclidean plane? Alternatively put, can a curvy (...

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212 views

### 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**

votes

**1**answer

120 views

### 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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**2**answers

421 views

### How many knots are there with hyperbolic volume less than a given constant

Are there any known upper bounds on:
$$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$
? I expect this grows at least exponentially in $M$, ...

**2**

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**0**answers

178 views

### Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that:
The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$.
For every quasiconvex subgroup $H \leq ...

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votes

**1**answer

661 views

### Any way to prove Prime Number Theorem using Hyperbolic Geometry? [closed]

The prime number theorem says that the density of prime numbers is inverse as the number of digits of $n$:
$$\displaystyle \frac{\{1 \leq k \leq n : \text{ prime } \}}{n} \approx \frac{1}{\log n}$$
...

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**0**answers

111 views

### 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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votes

**4**answers

785 views

### Geometry of the space of circles in the Euclidean plane

We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$.
It may even be possible to write an explicit formula ...

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**0**answers

82 views

### Is triple point intersection 'generic' in Teichmuller space?

Let $S$ be a hyperbolic surface of finite type and $\alpha,\beta$ be two closed curves. Consider $X$ to be the set of all those points $\chi$ in the Teichmuller space $\mathcal{T}(S)$ of $S$ such that ...

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**1**answer

80 views

### Additivity of simplicial volume

I have read for example in the introduction of http://arxiv.org/pdf/math/0506338v2.pdf about the property that if we glue hyperbolic manifolds with geodesic boundary consisting of tori along some tori,...

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**1**answer

85 views

### Equal-area projections of the hyperbolic plane [closed]

I'm aware of the projections of the hyperbolic plane at http://en.wikipedia.org/wiki/Hyperbolic_geometry#Models_of_the_hyperbolic_plane, but how would I project the hyperbolic plane onto a flat piece ...

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68 views

### Bi-Lipschitz classification of germs of conformal metrics at a singularity

First let me introduce some definitions.
By a germ of conformal metrics at a singularity, or simply a germ, I mean a conformal Riemannian metric $g$ defined on a punctured neighborhood $U$ of $0$ in ...

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**1**answer

247 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.
...

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**1**answer

201 views

### the paraboloid model for hyperbolic space [closed]

In Thurston's Three-Dimensional Geometry and Topology, he gives a recipe for a non-standard model for hyperbolic space which he calls the paraboloid model. I'd like to use the model to try out ...

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**3**answers

691 views

### Failure of Mostow rigidity in dimension 2

I am trying to understand why Mostow rigidity fails in dimension 2. More concretely, I have the following question:
(1) What is an example of a quasiisometry $f$ of the hyperbolic plane $\mathbb H^2$ ...

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vote

**1**answer

122 views

### What is the expression of first eigen function of Laplacian on Hyperbolic plane?

Let $\Delta$ be the Laplacian (a positive operator) on $H^2$ the hyperbolic plane. My question is what is the expression of the eigenfunction $\Delta f= f/4$? (say in the ploar coordiante)

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**1**answer

181 views

### 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 ...