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

It is well known that the homogeneous orthochronous proper Lorentzgroup is isomorphic to the proper motion group of the hyperbolic space. To each Lorentz boost \ {id} there corresponds in the hyperbolic space exactly one lineL such that fixes each of the two ends ofL . Furthermore has no fixed points but each plane containingL is fixed by . If we fix a pointo, then to each other pointa there is exactly one boosta ⁺ 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 P,b 0 andZ(b):= {x P |x+b=b+x}.Dedicated to H. Salzmann on the occasion of his 65th birthdaySupported by the NATO Scientific Affairs Division grant CRG 900103.