2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/188544We study geometry of two-dimensional models of conformal space-time based on the group of Moebius transformation. The natural geometric invariants, called cycles, are used to linearise Moebius action. Conformal completion of the space-time is achieved through an addition of a zero-radius cycle at infinity. We pay an attention to the natural condition of non-reversibility of time arrow in order to get a correct compactification in the hyperbolic case.8 pages,AMS-LaTeX, 18 PS figures; v2--small corrections; v3--add two coments on notations and multidimensional generalisationsMathematical Physics30G35, 22E46, 30F45, 32F45text