  We study expansive properties for the geodesic and horocycle flows on compact
Riemann surfaces of constant negative curvature. It is well-known that the
geodesic flow is expansive in the sense of Bowen-Walters and the horocycle flow
is positive and negative separating in the sense of Gura. In this paper, we
give a new proof of the expansiveness in the sense of Bowen-Walters for the
geodesic flow and show that the horocycle flow is positive and negative
kinematic expensive in the sense of Artigue as well as expansive in the sense
of Katok-Hasselblatt but not expensive in the sense of Bowen-Walters. We also
point out that the geodesic flow is neither positive nor negative separating.
