A geometric construction of the K-loop of a hyperbolic space

Abstract

It is well known that the homogeneous orthochronous proper Lorentzgroup Γ is isomorphic to the proper motion group of the hyperbolic space. To each Lorentz boost β ε Γ \ {id} there corresponds in the hyperbolic space exactly one lineL _β such that β fixes each of the two ends ofL _β . Furthermore β has no fixed points but each plane containingL _β is fixed by β. If we fix a pointo, then to each other pointa there is exactly one boosta ⁺ ε Γ such thatL _a+ is the line joiningo anda anda ⁺(o)=a. The set P of points of the hyperbolic space is turned in a K-loop (P, +) bya+b:=a ⁺(b). Each line of the hyperbolic space has the representationa+Z(b) wherea, b εP,b ≠ 0 andZ(b):= {x εP |x+b=b+x}.