  Aims: At intermediate redshift, galaxy groups/clusters are thought to impact
galaxies (e.g. their angular momentum). We investigate whether the environment
has an impact on the galaxies' angular momentum and identify underlying driving
physical mechanisms.
  Methods: We derive robust estimates of the stellar angular momentum using
Hubble Space Telescope (HST) images combined with spatially resolved ionised
gas kinematics from the Multi-Unit Spectroscopic Explorer (MUSE) for a sample
of ~200 galaxies in groups and in the field at z~0.7 drawn from the MAGIC
survey. Using various environmental tracers, we study the position of the
galaxies in the the angular momentum-stellar mass (Fall) relation as a function
of environment.
  Results: We measure a 0.12 dex (2sigma significant) depletion of angular
momentum for low-mass galaxies (M* < 10^10 Msun) in groups with respect to the
field. Massive galaxies located in dense environments have less angular
momentum than expected from the low-mass Fall relation but, without a
comparable field sample, we cannot infer whether this effect is mass- or
environmentally-driven. Furthermore, massive galaxies are found in the centre
of the structures and have low systemic velocities. The observed depletion of
angular momentum at low mass does not appear linked with the strength of the
over-density around the galaxies but it is strongly correlated with the
galaxies' systemic velocity normalised by the dispersion of their host group
and with their ionised gas velocity dispersion.
  Conclusions: Group galaxies seem depleted in angular momentum, especially at
low mass. Our results suggest that this depletion might be induced by physical
mechanisms that scale with the systemic velocity of the galaxies (e.g.
stripping or merging) and that such mechanism might be responsible for
enhancing the velocity dispersion of the gas as galaxies lose angular momentum.
