I am reading about Kleinian groups on Wikipedia and they show a picture of the Appolonian Gasket.

Let $\Gamma \subseteq \text{GL}_2(\mathbb{C})$ be a Kleinian group. The orbits of points under the Kleinian group accumulate at the sphere at infinity:
$$ \Omega = S^2_\infty - \Lambda$$ Here the shape we're interested is called the "limit set" and the complement of the limit set is called the "domain of discontinuity". Why is the boundary set called a "shere" and yet it's drawn as a subset of the "disc" here the union of countably many circles.

Could this just be the stereographic projection ? The Poincaré disc model describes $\mathbb{H}^2$ as a copy of the disc $D$ except they are saying the limit set is subset of the sphere $\Lambda \subset S^2$. This is a part of hyperbolic 3-space.

There are different models of hyperbolic space:



1 Answer 1


A Kleinian group $\Gamma \subseteq \text{SL}(2,\mathbb C)$ acts on the extended complex plane $\mathbb C^* = \mathbb C \cup \{\infty\}$ by fractional linear transformations. Using the identification $$\mathbb C = \{x+iy \mid x,y \in \mathbb R\} \approx \mathbb R^2 $$ that action extends further to the upper half space model of 3-dimensional hyperbolic space $$\mathbb H^3 = \{(x,y,z) \in \mathbb R^3 \mid z > 0\} $$ (the fractional linear action can be extended using quaternion notation $x + i \, y + j \, z + k \cdot 0$, which is described elsewhere on this site or in textbooks). The boundary of the upper half space model is literally $$\{(x,y,0) \in \mathbb R^3\} \cup \{\infty\} $$ but by using the identification $(x,y,0) \approx x+iy$ one may identify $\mathbb C^*$ with the boundary of $\mathbb H^3$.

With this in mind, the limit set $\Lambda\Gamma$ should be regarded as a subset of $\mathbb C^*$ and so should the domain of discontinuity $\Omega\Gamma = \mathbb C^* - \Lambda\Gamma$.

As you have surmised, we can indeed use 3-dimensional stereographic projection to identify $\mathbb C^* \approx S^2$ and so we can transfer our visualizations of limit sets and domains of discontinuity from $\mathbb C^*$ to $S^2$ if we so desire. By doing that we are choosing a different model of $\mathbb H^3$, namely the open 3-ball, known as the Poincaré ball model whose boundary is $S^2$. Of course this is all one dimension higher than the Poincaré disc model for $\mathbb H^2$ whose boundary is $S^1$.

The picture that you show of the Appolonian gasket should therefore be thought of not so much as a subset of the disc but instead as a subset of $\mathbb C^*$. From this point of view it is something of a coincidence that the Appolonian gasket is simultaneously the limit set of a certain Kleinian group (see lemma 3.4 here) and a subset of the Poincaré disc. (Regarding your title, that limit set is neither the whole disc in $\mathbb C^*$, nor the whole sphere $S^2$.)

To summarise, the various models for the 2-dimensional hyperbolic plane $\mathbb H^2$ (e.g. the upper half plane model; the Poincaré disc model; the Klein disc model) are not particularly appropriate for thinking about general Kleinian groups. Instead, for that purpose you should be learning about and thinking about the various models for the 3-dimensional hyperbolic space $\mathbb H^3$ (e.g. the upper half space model; the Poincaré ball model; the Klein ball model).


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