  We consider the region of closed timelike curves (CTC's) in three-dimensional
flat Lorentz spacetimes. The interest in this global geometrical feature goes
beyond the purely mathematical. Such spacetimes may be considered
lower-dimensional toy models of sourceless Einstein gravity or cosmology. In
particular, our interest in this note will be to find the set free of CTC's for
$E/<\gamma>$, where $E$ is modeled on Minkowski space and $\gamma$ is a
Poincar\'e transformation. We describe the set free of CTC's where $\gamma$ is
hyperbolic, parabolic and elliptic.
