# Tagged Questions

**2**

votes

**2**answers

360 views

### Dilogarithm, tetrahedrons, and hyperbolic space

The Bloch-Wigner function $D(z)$, gives the volume of a ideal tetrahedron in hyperbolic space $\mathbb H3$, where z is the cross-ratio (z1,z2,z3,z4) parametrizing the tetrahedron in $\mathbb CP1$.
...

**2**

votes

**2**answers

227 views

### Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb{H}^3$, has a boundary $\mathbb{CP}^1$.
A ideal tetrahedron in $\mathbb{H}^3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb{CP}^1$.
The four ...

**0**

votes

**0**answers

183 views

### Mapping from $\text{PSL}(2,\mathbb{R})$ to transformations of the hyperboloid

Let $F$ map points in $\mathbb{R}^3$ to points on the unit disk, $\Delta$, in the $xz$-plane (identified with $\mathbb{C}$) by projecting through $\Delta$ along lines that intersect at $(0,-1,0)$. ...

**4**

votes

**1**answer

274 views

### Arithmetic Fuchsian group

Dear all,
I have the following questions: Are all Fuchsian groups of signature $(0;2,2,2,\infty)$ arithmetic? What is known about the trace fields of these groups?
Best, K.

**12**

votes

**3**answers

629 views

### F→E→B bundle with B,E,F hyperbolic: possible?

It would be interesting to me obtain an answer to the following easy to state question:
Does there exist a (smooth) fibre bundle $\pi\colon E\rightarrow B$ with typical fibre $F$ such that $E$, $B$ ...

**3**

votes

**2**answers

390 views

### Capacity of Balls in Hyperbolic Space

Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ ...

**4**

votes

**6**answers

1k views

### Books for hyperbolic geometry ( surfaces ) with exercises ?

Hello, 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 ...

**2**

votes

**1**answer

646 views

### Hyperbolic structure on surfaces with boundary

I have following two questions
1) Let $S$ be a compact oriented surface with (non-empty) boundary. Also assume that the Euler characteristic of $S$ is negative (Thus, $S$ is not disk or annulus). ...

**16**

votes

**2**answers

2k views

### A question about the proof of Mostow rigidity

I have recently been studying a proof of Mostow rigidity (along the lines of Mostow's original argument), and I'm left a little confused about something. We start with an isomorphism $\alpha: \Gamma ...

**6**

votes

**1**answer

2k views

### Nice proof of the triangle inequality for the metric of the hyperbolic plane

I am writing something for the journal of the university on lorentz space, and I want to prove that the following definition of the hyperbolic distance on the upper sheet of the hyperboloid $v \cdot ...

**10**

votes

**3**answers

804 views

### Tetrahedra with prescribed face angles

I am looking for an analogue for the following 2 dimensional fact:
Given 3 angles $\alpha,\beta,\gamma\in (0;\pi)$ there is always a triangle with these prescribed angles. It is ...

**6**

votes

**4**answers

1k views

### Finding Constant Curvature Metrics on Surfaces without full power of Uniformization

(I rewrote this question, hopefully it's more clear now. It's still the same question, but I reordered its parts.)
Let S be a surface (possibly non-compact, but no boundary). It seems that there are ...

**37**

votes

**6**answers

2k views

### What do Weierstrass points look like?

As somebody who mostly works with smooth, real manifolds, I've always been a little uncomfortable with Weierstrass points. Smooth manifolds are totally homogeneous, but in the complex category you ...