  In the first part of this paper we consider expanding vacuum cosmological
spacetimes with a free $T^N$-action. Among them, we give evidence that Gowdy
spacetimes have AVTD (asymptotically velocity term dominated) behavior for
their initial geometry, in any dimension. We then give sufficient conditions to
reach a similar conclusion about a $T^2$-invariant four dimensional nonGowdy
spacetime. In the second part of the paper we consider vacuum cosmological
spacetimes with crushing singularities. We introduce a monotonic quantity to
characterize Kasner spacetimes. Assuming scale-invariant curvature bounds and
local volume bounds, we give results about causal pasts.
