2
votes
2answers
381 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
2answers
236 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
0answers
189 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
1answer
283 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
3answers
631 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
2answers
394 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 $\varphi$ ...
5
votes
6answers
2k 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
1answer
672 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
2answers
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
1answer
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
3answers
826 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
4answers
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
6answers
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 ...