Semigroups in Möbius and Lorentzian Geometry

The Möbius semigroup studied in this paper arises very naturally geometrically as the (compression) subsemigroup of the group of Möbius transformations which carry some fixed open Möbius ball into itself. It is shown, using geometric arguments, that this semigroup is a maximal subsemigroup. A detailed analysis of the semigroup is carried out via the Lorentz representation, in which the semigroup resurfaces as the semigroup carrying a fixed half of a Lorentzian cone into itself. Close ties with the Lie theory of semigroups are established by showing that the semigroup in question admits the structure of an Ol'shanskii semigroup, the most widely studied class of Lie semigroups.