Topology and dynamics of compact plane waves

We study compact locally homogeneous plane waves. Such a manifold is a quotient of a homogeneous plane wave 𝑋 by a discrete subgroup of its isometry group. This quotient is called standard if the discrete subgroup is contained in a connected subgroup of the isometry group that acts properly cocompactly on 𝑋. We show that compact quotients of homogeneous plane waves are “essentially” standard; more precisely, we show that they are standard or “semi-standard”. We find conditions which ensure that a quotient is not only semi-standard but even standard. As a consequence of these results, we obtain that the flow of the parallel lightlike vector field of a compact locally homogeneous plane wave is equicontinuous.