  Post-starburst (E+A or k+a) spectra, characterized by their exceptionally
strong Balmer lines in absorption and the lack of emission lines, belong to
galaxies in which the star formation activity ended abruptly sometime during
the past Gyr. We perform a spectral analysis of galaxies in clusters, groups,
poor groups and the field at z=0.4-0.8 based on the ESO Distant Cluster Survey.
The incidence of k+a's at these redshifts depends strongly on environment.
K+a's reside preferentially in clusters and, unexpectedly, in a subset of the
sigma = 200-400 km/s groups, those that have a low fraction of [OII] emitters.
In these environments, 20-30% of the recently star-forming galaxies have had
their star formation activity recently truncated. In contrast, there are
proportionally fewer k+a's in the field, the poor groups and groups with a high
[OII] fraction. The incidence of k+a galaxies correlates with the cluster
velocity dispersion: more massive clusters have higher proportions of k+a's.
Spectra of dusty starburst candidates, with strong Balmer absorption and
emission lines, present a very different environmental dependence from k+a's.
They are numerous in all environments at z=0.4-0.8, but they are especially
numerous in all types of groups, favoring the hypothesis of triggering by a
merger. Our observations are consistent with previous suggestions that cluster
k+a galaxies are observed in a transition phase as massive S0 and Sa galaxies,
evolving from star-forming later types to passively evolving early-type
galaxies. The correlation between k+a fraction and cluster sigma supports the
hypothesis that k+a galaxies in clusters originate from processes related to
the intracluster medium, while several possibilities are discussed for the
origin of the k+a frequency in low-[OII] groups.(abr.)
