  We present the goals, design, and first results of the MUSE Ultra Deep Field
(MUDF) survey, a large programme using the Multi Unit Spectroscopic Explorer
(MUSE) instrument at the ESO Very Large Telescope. The MUDF survey is
collecting ~ 150 hours on-source of integral field optical spectroscopy in a
1.5 x 1.2 square arcmin region which hosts several astrophysical structures
along the line of sight, including two bright z ~ 3.2 quasars with close
separation (~ 500 kpc). Following the description of the data reduction
procedures, we present the analysis of the galaxy environment and gaseous
properties of seven groups detected at redshifts 0.5 < z < 1.5, spanning a
large dynamic range in halo mass, log(Mh/Msun) ~ 11 - 13.5. For four of the
groups, we find associated MgII absorbers tracing cool gas in high-resolution
spectroscopy of the two quasars, including one case of correlated absorption in
both sightlines at distance ~ 480 kpc. The absorption strength associated with
the groups is higher than what has been reported for more isolated galaxies of
comparable mass and impact parameters. We do not find evidence for widespread
cool gas giving rise to strong absorption within these groups. Combining these
results with the distribution of neutral and ionised gas seen in emission in
lower-redshift groups, we conclude that gravitational interactions in the group
environment strip gas from the galaxy haloes into the intragroup medium,
boosting the cross section of cool gas and leading to the high fraction of
strong MgII absorbers that we detect.
