  We investigate the emission-line properties of galaxies with red rest-frame
colors using spectra from SDSS DR4. Emission lines are detected in more than
half of the red galaxies. We focus on the relationship between two emission
lines commonly used as star formation rate indicators: Ha 6563 and [OII] 3727.
There is a strong bimodality in [OII]/Ha ratio in the full SDSS sample which
closely corresponds to the bimodality in rest-frame color. Nearly all of the
line-emitting red galaxies have line ratios typical of various types of AGN --
most commonly LINERs, a small fraction of transition objects and, more rarely,
Seyferts. Only ~6% of red galaxies display star-forming line ratios. A straight
line in the [OII]-Ha equivalent width plane separates LINER-like galaxies from
other categories. Quiescent galaxies with no detectable emission lines and
LINER-like galaxies combine to form a single, tight red sequence in
color-magnitude-concentration space. [OII] EWs in LINER- and AGN-like galaxies
can be as large as in star-forming galaxies. Thus, unless objects with
AGN/LINER-like line ratios are excluded, [OII] emission cannot be used directly
as a proxy for star formation rate. Lack of [OII] emission is generally used to
indicate lack of star formation when post-starburst galaxies are selected at
high redshift. Our results imply, however, that these samples have been cut on
AGN properties as well as star formation, and therefore may provide seriously
incomplete sets of post-starburst galaxies. Furthermore, post-starburst
galaxies identifed in SDSS by requiring minimal Ha EW generally exhibit weak
but nonzero line emission with ratios typical of AGNs; few of them show
residual star formation. This suggests that most post-starbursts may harbor
AGNs/LINERs.
