A Survey of Möbius Groups

Classical Fuchsian and Kleinian groups are offspring of the theory of functions of one complex variable. Like modern complex analysis they were born in the nineteenth century. They are groups of conformal homeomorphisms of the Riemann sphere identified either with the extended complex plane $$ \bar C$$ = C ∪ {∞} or with the 2-sphere S2 = {x ∈ R3 : ❘x❘ = 1}, and with discontinuous action on an open nonempty set. Poincaré [Po] had already found out that there is a natural extension of the group action to the upper half-space H3 = {(x1,x2,x3) ∈ R3 : X3 > 0}. Poincaré’s extension was based on the fact that any conformal homeomorphism of C can be represented as a composition of inversions (also called reflections) on spheres, the prototypical inversion being the inversion X ü X/❘X❘2 of the unit sphere. Obviously, an inversion is extendable to H3 (or in fact to Xn = Xn ∪ {∞} with arbitrary dimension n). In this manner it is possible to extend the action of a Kleinian group to H3.