# Tagged Questions

**2**

votes

**1**answer

154 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 ...

**6**

votes

**0**answers

178 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

**1**answer

344 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

**2**answers

220 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

**0**answers

113 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

**1**answer

205 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

**2**answers

142 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

**3**answers

264 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

**0**answers

282 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 ...

**2**

votes

**2**answers

220 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

**1**answer

71 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

**1**answer

466 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

**1**answer

281 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

**1**answer

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

**3**answers

717 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"[*] ...

**9**

votes

**6**answers

669 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 ...