# Tagged Questions

Questions on hyperbolic geometry, the geometry on manifolds with negative curvature.

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### Limit set of Kleinian group

Let $\Gamma \subset PSL_2 (\mathbb{C})$ a Kleinian group coming from a discrete faithful representation $\rho : \pi_1(M) \to PSL_2 (\mathbb{C})$ of the fundamental group of a closed connected ...
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### Hyperbolic metric of arbitrary curvature.

I've been trying to find this online, in books, etc, but I can never find the expression for the metric on the unit disk $$\mathbb{D} = \{ z \in \mathbb{C} : |z| < 1 \}$$ that has constant ...
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### Distance from point to line segment in Poincaré disk model

I'm trying to build a geometric datastructure in hyperbolic space. For that purpose, I'm using the Poincaré disk model. The distance between two points can be calculated with the hyperbolic law of ...
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### Constructing The Cayley Graph and quasi-isometry to $\mathbb{Z}$

If we have a group $G$ defined by: $G=\langle a,b\mid b^2=1\rangle$ then I first need to construct the cayley graph of this, now I think that this is going to look like the "telephone pole" metric ...
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### Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
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### Showing geodesics in $\mathbb{H}^2$ have unique common perpendicular or common endpoint

If we let $\mathbb{H}^2$ be the hyperbolic plane and we let $\gamma_1,\gamma_2$ be geodesics which do not intersect. I have a question which asks me to show that either $\gamma_1$ and $\gamma_2$ have ...
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### Hyperbolic metric spaces

I'm trying to prove a simple proposition wich is in Burago's "A Course in Metric Spaces" (Exercise $8.4.5$, p.$287$). Before exposing my problem, let me give some definitions. A metric space $(X,d)$ ...
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### oriented surface of genus g and m punctures

Let $S_{g,m}$ be an oriented surface of genus g and m punctures, what's the condition to ensure $S_{g,m}$ is hyperbolic? If $g\ge 2$, I know it is hyperbolic, how about g=0 and g=1? Thanks in advance. ...
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### Definition of complex hyperbolic geometry

I am trying to read about complex hyperbolic geometry.But I couldnot find a basic definition for it. Is it just the special case of hyperbolic geometry where we work with complex numbers in the model. ...
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### Are there other models for 2 dimensional hyperbolic geometry?

I was a bit browsing the internet for models for (2-dimensional) hyperbolic geometry. and realised that besides the well known Poincare half plane model Poincare disk model Beltrami-Klein disk ...
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### Measuring distance on the Poincare disk

I've seen several different ways to measure distance on the Poincare disk i.e Riemann metric/manifold (which I don't understand). However the method we're taught is using $\tanh^{-1}$ and complex ...