Using computations motivated by theoretical physics, Ferrara et al. (1973) introduced the semigroup H , defined through a set of inequalities. Without identification H as a compression semigroup, they established the maximality of H ¯ ¯ . Using the definition as a compression semigroup, maximality reduces to checking what happens when adding a new fractional transformation g to H ¯ ¯ . The idea of the proof depends on considering the positions of the two discs g ( D ) and D . In the key cases, either one disc contains the other or they are disjoint. In the simplest cases, g is the inverse of a scaling transformation or g ( z ) = − − 1 / z . In either case g and H generate an open neighbourhood of 1 and hence the whole of SL(2,C)