Tagged Questions
2
votes
0answers
55 views
Reference Request: Regge Symmetry “Angle-Edge” Duality
A tetrahedron in hyperbolic 3-space can be defined (up to isometry) by the measures of its dihedral angles, $(a, b, c, a^\prime, b^\prime, c^\prime)$, with $a$, $b$, $c$ along edges that meet at a ...
0
votes
0answers
21 views
Enlightening explanation of a theorem of Zimmert's
I'd like to know wether anyone has ever read an enlightening explanation (e.g. with geometric argument) of the following paper:
Zimmert, R.
Zur $SL_2$ der ganzen Zahlen eines imaginär-quadratischen ...
3
votes
0answers
86 views
Axis of the product of two loxodromic isometries
Suppose that $X$ and $Y$ are two loxodromic isometries of the hyperbolic space and that the product $XY$ is also a loxodromic element.
We consider the axes of these three elements. I'd like to know if ...
2
votes
1answer
58 views
Hyperbolic spheres in the Poincare half-plane and fractional linaear transformations
Let $\mathbb{H}$ be the Poincare upper half-plane, seen as a
Riemannian manifold with the metric $$\frac{dx^2+dy^2}{y^2}.$$
Moreover, we consider the action of $\text{SL}_2(\mathbb{R})$ on
...
3
votes
0answers
54 views
reference request: “p-adic” presentation of surfaces
On several occasions I heart about the following result:
For "certain" lattices $\Lambda$ in $SL_2(\mathbb{R})$, and almost any prime $p$ there exists a lattice $\Gamma$ in $SL_2(\mathbb{R})\times ...
3
votes
2answers
65 views
Real tree and hyperbolicity
I seek a proof of the following result due to Tits:
Theorem: A path-connected $0$-hyperbolic metric space is a real tree.
Do you know any proof or reference?
0
votes
1answer
49 views
Polynomial behavior on hyperbolic plane
More a reference request / more information.
I was reading some websites about hyperbolic geometry and got to thinking about how would polynomials $(x^2-2)$ behave in such a geometry.
So, I need ...
2
votes
1answer
40 views
Character Varieties- reference request
I will start learning about character varieties. I need to learn about Teichmuller spaces and how to consider them as components of "some character variety".
Can someone recommend some textbooks or ...
1
vote
1answer
51 views
Limit sets of representations of once-punctured torus groups and circle packings
Let $\rho\colon\pi_1(T_1)\to PSL(2,\mathbb{C})$ be a faithful representation of the fundamental group of a once-punctured torus. If both the components of the convex core in the quotient manifold are ...
1
vote
1answer
147 views
A book to study about hyperbolic plane, hyperbolic translations, etc.
In this paper, page $6$, the authors state the following:
The translations of the hyperbolic plane are defined as products of
two central symmetries; the set of hyperbolic translations forms a
...
3
votes
1answer
157 views
Thrice-punctured sphere
This claim is made in the book Quantum Triangulations (eds.: Carfora, Marzuoli), p.45:
the thrice-punctured sphere is the largest subdomain of $\mathbb{S}^2$ supporting a hyperbolic metric.
I ...
2
votes
1answer
86 views
A Kleinian group has the same limit set as its normal subgroups'
It should be well known that a Kleinian group and all its normal (non-elementary) subgroups have the same limit set. Do you know any book/article where I could find the proof?
Thank you.
7
votes
1answer
199 views
Is it possible to deduce a model for hyperbolic geometry from a synthetic set of axioms a la Euclid/Hilbert/Tarski?
Motivation
I learned from Emil Artin's book Geometric Algebra that the standard incidence axioms of affine geometry (two points determine a unique line, parallel postulate, no three collinear points ...
3
votes
1answer
246 views
Models of hyperbolic geometry
Wikipedia states the following:
[The Poincaré half-plane model of hyperbolic geometry] is named after Henri Poincaré, but originated with Eugenio Beltrami, who used it, along with the Klein model ...
