Kleinian and Fuchsian groups

Let $ G$ be a group of Möbius transformations. We say that $ G$ acts properly discontinuously at a point $ z$ of $ \widehat{\mathbb{C}}$ if there exists an open neighbourhood $ U$ of $ z$ such that the subset of $ G$ defined by

$\displaystyle \left\{g\in G/ g(U) \cap U \neq \emptyset \right\}

is finite. Clearly if $ G$ acts properly discontinuously at $ z$ then $ G$ will act properly discontinuously at all points of $ U$, so we have that the set of points where $ G$ acts properly discontinuously is an open set of the Riemann sphere. We call this set the set of discontinuity of $ G$ and denote it by $ \Omega(G)$ (or simply $ \Omega$). Its complement on the Riemann sphere is called the limit set of $ G$, $ \Lambda(G)$ (or just $ \Lambda$).

Definition 1.9   A group of Möbius transformations is called Kleinian if $ \Omega(G) \neq \emptyset$.

With the topology of Möbius transformations explained above we have the following result.

Prop 1.10   A Kleinian group is discrete subgroup of $ \mathrm{PLS}(2,{\mathbb{C}})$.

Remark: frequently the term Kleinian is used for a discrete groups of Möbius transformations, even if $ \Omega(G) = \emptyset$ (Thurston's definition). This is motivated by the fact that a discrete group of Möbius transformations acts properly discontinuously on the $ 3$-dimension hyperbolic space. See proposition 1.17 below for a similar result for Fuchsian groups.

Prop 1.11   If $ G$ is a Kleinian group then $ \Lambda$ is closed, $ G$-invariant, nowhere dense subset of $ \widehat{\mathbb{C}}$.

An interesting property of Kleinian groups is that their limit sets have at most two points or are uncountable (a similar property holds for iteration of rational functions on the Riemann sphere).

Prop 1.12   If $ G$ is a Kleinian group then $ \Lambda$ consists of zero, one, two or uncountably many points. In the latter case, $ \Lambda$ is perfect.

A Kleinian group whose limit set consists of at most two points is called an elementary group.

Prop 1.13   For a Kleinian group $ G$ we have that the Riemann sphere is the disjoint union of $ \Omega$ and $ \Lambda$.

Elementary groups appear as groups of deck transformations of branched coverings of certain Riemann surfaces. We can list them all, but before that we need to set up some notation. Let $ \tilde{X}$ and $ X$ be two Riemann surfaces and $ \pi:\tilde{X} \to X$ a mapping. We say that $ \pi$ is a branched covering if at every point $ p$ of $ X$ we have that $ \pi$ is either a covering in the usual sense (called regular covering by some authors), or $ \pi$ is locally like $ z \mapsto z^k$ for some positive integer $ k > 1$. In the latter case we say that $ p$ is a ramification point and $ k$ is called the ramification value of $ \pi$. We also require that the set of points with ramification is a discrete subset of $ X$. If $ X$ is a compact surface of genus $ g$, $ \tilde{X}$ is simply connected, and $ \pi$ is ramified over $ p_1,\ldots,p_n$, with ramification values $ \nu_1,\ldots,\nu_n$, we say that $ X$ has signature $ (g,n;\nu_1,\ldots,\nu_n)$. For notation purposes we allow the symbol $ \infty$ in a signature, to include punctures. A puncture is just a missing point of $ X$ (or, more formally, a domain on $ X$ conformally equivalent to the unit disc minus the origin). We include the punctures of a surface in its signature with the symbol $ \infty$; for example, the complex plane is equal to the Riemann sphere with one puncture, so its signature is $ (0,1;\infty)$, and the complex plane minus the origin has signature $ (0,2;\infty,\infty)$. With this notation we can describe all elementary groups.

Suppose $ G$ is an elementary, non trivial group with no limit points. Then $ G$ is the deck transformations of a covering $ \widehat{\mathbb{C}}
\to \widehat{\mathbb{C}}$, branched over two or three points. The possible signatures of such covering are:

$\displaystyle (0,2;\nu,\nu), \hspace{6pt} \nu > 1;\hspace{12pt}
(0,3;2,2,\nu), \hspace{6pt} \nu > 1;\hspace{12pt}
(0,3;2,3,\nu), \hspace{6pt} \nu=3,4,5.

If $ G$ has one limit point then $ G$ (after conjugation) acts properly on the complex plane; that is, it is the deck transformation group of a (possibly branched) covering of the form $ {\mathbb{C}} \to X$. We have the following possible signatures:

$\displaystyle (1,0); \hspace{6pt} (0,2;\infty,\infty); \hspace{6pt}
(0,3;3,3,3); \hspace{6pt} (0,3;2,4,4); \hspace{6pt} (0,3;2,2,2,2).

The first signature corresponds to a torus and the second one to the complex plane minus one point; the other signatures correspond to different branched coverings of the complex plane or the Riemann sphere.

If $ G$ has two limit points we have that it acts (after conjugation) on $ {\mathbb{C}} \backslash \{0\}$; the corresponding coverings are those of a torus (signature $ (1,0)$) and a sphere with four ramification points of order $ 2$ (signature $ (0,4;2,2,2,2)$).

A Kleinian group is called Fuchsian if it leaves a disc $ U$ invariant. Here by a disc (in the Riemann sphere) we understand some circular disc or a half-plane. The boundary of $ U$ is called the circle at infinity of $ G$. We say that the group is of the first kind if every point of the circle at infinity is a limit point ( $ \Lambda = U$); otherwise the group is called of the second kind ( $ U \subset
\Lambda$). By conjugation in $ \mathrm{PSL}(2,{\mathbb{C}})$ we can assume, when necessary, that a Fuchsian group leaves the upper half plane invariant; in that case $ G$ can be considered as a subgroup of $ \mathrm{PSL}(2,{\mathbb{R}})$.

Prop 1.14   Let $ G$ be a Fuchsian group of the second kind. Then $ \Lambda$ is nowhere dense in the circle at infinity.

We set the convention $ \frac{1}{\infty} = 0$ and for a signature $ (g,s;n_1,\ldots,n_s)$ we put $ \chi = 2g - 2 + n - \sum_{j=1}^s
\frac{1}{n_j}$ (the negative of the Euler characteristic of the covering). Then we have that for an elementary group with no limit points, $ \chi < 0$, while for an elementary group with one or two limit points we get $ \chi = 0$. The next result says that if a signature satisfies $ \chi > 0$ then it is the signature of a (possibly branched) covering defined on the upper half plane.

Prop 1.15   Let $ \sigma$ be a signature satisfying $ \chi > 0$. Then there exists a Fuchsian group of the first kind $ G$, acting on the upper half plane, and a Riemann surface $ X$, such that $ {\mathbb{H}} \to X$ is a branched covering with the given signature and covering transformation group $ G$.

Theorem 1.16   Let $ G$ be a non-elementary group in which $ tr^2(g) \geq 0$ for all $ g \in G$. Then $ G$ is Fuchsian.

Prop 1.17   Let $ G$ be a subgroup of $ \mathrm{PSL}(2,{\mathbb{R}})$. Then the following are equivalent:
(i) $ G$ is acts properly discontinuously on the upper half;
(ii) $ G$ is discrete.

This result is the motivation for the definition of a Kleinian group as a discrete group of Möbius transformations: one such group will act properly discontinuously on the $ 3$-dimensional hyperbolic sphere (whose boundary is the Riemann sphere). The conclusion of the proposition is not true if we remove the condition of the group being Fuchsian. For example the group consisting of all Möbius transformations with entries in the Gaussian integers, the Picard group, is discrete, but it does not act properly discontinuously at any point of the Riemann sphere.

Pablo Ares Gastesi 2005-08-31