Relativistic modifications to wave dispersion near the electron gyrofrequency are known to be important for sufficiently energetic electrons. In general, if the characteristic momentum of the energetic particles (p0/m0c) is greater than the ratio of the plasma frequency to the gyrofrequency (&ohgr;p/&OHgr;e), wave propagation can be markedly different than the classical cold plasma limit. We investigate the wave dispersion in a weakly relativistic electron plasma, specifically in the context of the generation of auroral kilometric radiation (AKR). To do this, we have used a simple electron distribution consisting of a ring of energetic electrons at constant perpendicular momentum together with a stationary cold component. It is found that for such an electron distribution, where the two electron components have different gyrofrequencies due to relativistic effects, a new wave mode is introduced. This wave is trapped between the two electron gyrofrequencies. We find that the group velocity for such a wave is small; convective growth lengths for the wave are calculated to be of the order of 1 km. Consequently, ray propagation in a simple model for the density cavity present on auroral zone field lines is studied. We find that the ray paths tend to bring waves to altitudes where the growth rate maximizes; at these altitudes the waves propagate primarily across the ambient field. This could result in substantial amplification of the wave, since damping is minimized, and the waves may undergo multiple transitions across the auroral arc. However, the wave is trapped and must undergo mode conversion to escape from the auroral arc. Since the wave polarization is found to be very similar to the R-Z mode polarization, we deduce that coupling to the R-X mode at the edge of the auroral arc may be reasonably efficient.