  We 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.
