Relations between the half turns of the hyperbolic plane

Let R_a 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 R_aR_b = R_cR_d. We study the group G whose generating set is {R_a:a∈H} and whose defining relations are the ones mentioned above together with the relations R²_a = 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.