Fuchsian Group - General Definition

General Definition

A linear fractional transformation defined by a matrix from PSL(2,C) will preserve the Riemann sphere P1(C) = C ∪ ∞, but will send the upper-half plane H to some open disk Δ. Conjugating by such a transformation will send a discrete subgroup of PSL(2,R) to a discrete subgroup of PSL(2,C) preserving Δ.

This motivates the following definition of a Fuchsian group. Let Γ ⊂ PSL(2,C) act invariantly on a proper, open disk Δ ⊂ C ∪ ∞, that is, Γ(Δ) = Δ. Then Γ is Fuchsian if and only if any of the following three properties hold:

1. Γ is a discrete group (with respect to the standard topology on PSL(2,C)).
2. Γ acts properly discontinuously at each point z ∈ Δ.
3. The set Δ is a subset of the region of discontinuity Ω(Γ) of Γ.

That is, any one of these three can serve as a definition of a Fuchsian group, the others following as theorems. The notion of an invariant proper subset Δ is important; the so-called Picard group PSL(2,Z) is discrete but does not preserve any disk in the Riemann sphere. Indeed, even the modular group PSL(2,Z), which is a Fuchsian group, does not act discontinuously on the real number line; it has accumulation points at the rational numbers. Similarly, the idea that Δ is a proper subset of the region of discontinuity is important; when it is not, the subgroup is called a Kleinian group.

It is most usual to take the invariant domain Δ to be either the open unit disk or the upper half-plane.