  We present a solution for the geodesic motion in G\"odel's universe that
provides a particular proof of Hawking's chronology protection conjecture in
three-dimensional gravity theory. The solution is based upon the fact that the
group of the automorphisms of the Heisenberg motion group H1\timesU(1), modulo
discrete sub-group Z, act isometrically on the boundary of the hyperbolic
three-dimensional manifold. Closed timelike curves do not exist due to the
presence of a closed Cauchy-Riemann surface for chronology protection, with two
mirror symmetric sets of helicoidal self-similar modules inside. The present
solution is isometrically equivalent to a cylindrical gravitational
monochromatic wave front.
