3
votes
1answer
195 views

Dirichlet polyhedra for hyperbolic manifolds

Let $H$ be a simply-connected, complete space of constant negative curvature, that is, a hyperbolic space, $\Gamma$ a discrete group of isometries, and and $M=H/\Gamma$ its quotient space; we assume ...
16
votes
2answers
371 views

Is every elementary absolute geometry Euclidean or hyperbolic?

Absolute geometry is any one that satisfies Hilbert's axioms of plane geometry without the axiom of parallels. It is well-known that it is either the Euclidean or a hyperbolic plane. For an elementary ...
8
votes
1answer
433 views

Is there a straightedge and compass construction of incommensurables in the hyperbolic plane?

In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the ...
1
vote
0answers
55 views

Coarse geometry of minimal surfaces in non-positively curved manifolds

Let $X$ be a simply-connected Riemannian manifold of non-positive curvature and $S\subset X$ be a complete minimal surface. (You can basically image $X$ as a ball and $S$ as an embedded disk whose ...
2
votes
1answer
170 views

Model of hyperbolic geometry with finite number of parallel line

Does there exist a model of hyperbolic geometry such that only finite number of distinct parallel lines through a point which does not intersect given line? Edit (Misha): I usually do not edit other ...
7
votes
0answers
189 views

Right-angled polytopes

%This question is motivated by the little discussion here at the bottom. The following thing are known about hyperbolic right-angled polytopes: Compact hyperbolic right-angled polytopes do not ...
7
votes
1answer
347 views

Do elements of the fundamental group give rise to isometries

Let $X$ be a complex algebraic variety, and let $\tilde X\to X$ be its universal cover. Suppose that there exists a Kahler-Einstein metric on $\tilde X$. Note that $\pi_1(X) \subset Aut(\tilde X)$. ...
6
votes
2answers
235 views

How close can closed geodesics be?

A consequence of the famous Jørgensen inequality is that there is a lower bound for the distance between closed geodesics in hyperbolic three-manifolds: for any $R>0$ there is a c>0 such ...
3
votes
0answers
129 views

The distance between two farthest points on the Bolza surface?

The Bolza surface $M$ is the closed hyperbolic surface of genus $2$ that can be obtained by identifying the opposite sides of the regular octagon in $\mathbb{H}^2$. What two points on $M$ are ...
5
votes
1answer
225 views

Standard (special) spines and hyperbolic structure on 3-manifolds

My question relates to constructing angled triangulations or hyperbolic triangulations for $3$--manifolds. Briefly, an angle triangulation can be considered as an assignment of a real number (called ...
4
votes
2answers
153 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 ...
2
votes
3answers
270 views

Mid point with set square?

Is it possible to construct the midpoint of a segment in the hyperbolic plane using the set square only? With the set square one can draw the line through the given two points and drop the ...
6
votes
0answers
295 views

A conjecture of Thurston and possibly Weeks too

What is the status of the following conjecture: "... [w]hen the shortest simple closed geodesics are repeatedly removed from any complete hyperbolic 3-manifold of finite volume, eventually one ...
1
vote
2answers
242 views

Combination theorems for discrete subgroups of isometry groups

Maskit's combination theorem says: if $M=M_1\cup_\Sigma M_2$ is a union of hyperbolic 3-manifolds $M_1=\Gamma_1\backslash H^3, M_2=\Gamma_2\backslash H^3$ along a surface $\Sigma$, and if the limit ...
1
vote
1answer
72 views

Log-nonexpansive functions: terminology and references

During my recent work in the optimization of positive valued functions, the following class of functions proved to be exceptionally important. (Defn.). Let $h: (0,\infty) \to (0,\infty)$ be ...
0
votes
1answer
482 views

Is there a similar formula in spherical and hyperbolic geometry as Euclidean Geometry? [closed]

In an Euclidean plane, we know that the area of a triangle is determined by the length of base and the height, i.e., $$ S_{\Delta}=\frac{1}{2}a.h, $$ where $a$ is the length of base and the $h$ is ...
2
votes
1answer
296 views

Straight line on the Poincare disk hitting points almost everywhere

Consider the tiling of the Poincare disk $\mathbb{D}$ by identified octagons (i.e., representing a torus with genus 2). Suppose inside each octagon is a subset A such that the octagon minus A is a ...
10
votes
1answer
1k views

Pythagorean Theorem for Right-Corner Hyperbolic Simplices?

My answer to the "Favorite Equations" question was the Pythagorean Theorem for Right-Corner Tetrahedra: Euclidean: $A^2+B^2+C^2=D^2$ Hyperbolic: ...
7
votes
3answers
750 views

What are trig classes like within a universe that's “noticeably” hyperbolic?

[I want to think that this question has an answer, but it may be more a "community wiki" discussion. Feel free to re-tag.] What are trig classes like within a universe that's "noticeably"[*] ...
15
votes
6answers
966 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 ...
9
votes
6answers
696 views

How to smootly interpolate between möbius transformations?

If you have two Möbius transformations represented as: $f(z) = \frac{az + b}{cz + d}$ $g(z) = \frac{pz + q}{rz + s}$ where $a, b, c, d, p, q, r, s, z \in \mathbb{C}$ Is it possible to derive a ...