Relations between the half turns of the hyperbolic plane

AbstractLet Ra denote the half turn about the point a of the hyperbolic plane H. If the points a, b, c, d lie on the same line and the pair (c, d) is obtained from the pair (a, b) by a translation, then we have RaRb = RcRd. We study the group G whose generating set is {Ra:a∈H} and whose defining relations are the ones mentioned above together with the relations R2a = 1. We show that G can be made into a Lie group, G has two connected components, and its identity component G0 is the universal covering group of PSL2(R). In particular, it follows that all relations between the half turns in PSL2(R) follow from the abovementioned relations and a single additional relation of length five