Studying the flow of baryons into and out of galaxies is an important part of understanding the evolution of galaxies over time. We present a detailed case study of the environment around an intervening Ly $alpha$ absorption line system at $z_{rm abs} = 0.633$, seen towards the quasar J0423$-$0130 ($z_{rm QSO} = 0.915$). We detect with ALMA the $^{12}$CO(2--1), $^{12}$CO(3--2) and $1.2$~mm continuum emission from a galaxy at the redshift of the Ly $alpha$ absorber at a projected distance of $135$ kpc. From the ALMA detections, we infer ISM conditions similar to those in low redshift Luminous Infrared Galaxies. DDT MUSE integral field unit observations reveal the optical counterpart of the $^{12}$CO emission line source and three additional emission line galaxies at the absorber redshift, which together form a galaxy group. The $^{12}$CO emission line detections originate from the most massive galaxy in this group. While we cannot exclude that we miss a fainter host, we reach a dust-uncorrected star-formation rate (SFR) limit of > $0.3 text{M}_{odot} text{ yr}^{-1}$ within $100$ kpc from the sightline to the background quasar. We measure the dust-corrected SFR (ranging from $3$ to $50$ M$_{odot}$ yr$^{-1}$), the morpho-kinematics and the metallicities of the four group galaxies to understand the relation between the group and the neutral gas probed in absorption. We find that the Ly $alpha$ absorber traces either an outflow from the most massive galaxy or intra-group gas. This case study illustrates the power of combining ALMA and MUSE to obtain a census of the cool baryons in a bounded structure at intermediate redshift.
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