tokens: inferred from the shell modeling reported in Sect. 3.1 and Fig. 11 must be considered as strict lower limits. This is because the emerging spectrum is strongly weighted by sightlines with the lowest column densities, given that Ly α escapes through the paths of least resistance (e.g., Eide et al. 2018 ), generally orthogonal to the stream velocities in case of infall ( Gronke et al. 2017 ). Nevertheless, we note that an alternative scenario suggested by recent modeling is that radiation is emitted through the cooling of mixed gas occurring at the boundary between the phases ( Mandelker et al. 2019 ; Cornuault et al. 2018 ). In that case, Ly α radiation would originate from the combined dissipation of the kinetic energy of the stream and the thermal energy from the mixed gas that cools down ( Gronke Oh 2020a ). This scenario, and our calculations of gas mass flows in Sect. 3.1 , also imply a total cold gas mass present at each moment in the diffuse streams of the order of 10 11 M ⊙ (to be compared to the lower limit of ≈10 8 M ⊙ of neutral hydrogen). Is that a reasonable gas mass to be diffused into the intra-group gas? On the one hand, this is comparable to or smaller than the ISM mass of all group galaxies combined, and less than 10% of all baryons expected to be in the group given its DM estimate and assuming a universal baryon fraction. Assuming pressure equilibrium, the flowing gas would need to be confined in clouds with quite a small volume filling factor, of order 10 −3 to 10 −5 , which is qualitatively similar to what is discussed in theoretical works (e.g., Cornuault et al. 2018 ; McCourt et al. 2018 ; Gronke Oh 2020b ). This is required to reach a high-enough density to compensate for the much larger temperature of the hot medium. Such dense gas clouds would produce copious recombination Ly α photons following photoionization, if a source emitting enough ionizing photons towards this medium were to be present. This does not appear to be the case for RO-1001, at least given the evidence presented here. 5. Summary and conclusions The main findings of this work can be summarized as follows: RO-1001 is a group at z = 2.91 defined by four galaxies with stellar masses above 10 11 M ⊙ , originally selected as an overdensity of faint radio sources (12 σ excess) over a small, 10′′ region. The group contains an ongoing SFR of 1200 M ⊙ yr −1 mostly spread among three ALMA-detected, IR-luminous sources characterized by very compact IR sizes, for which we reported CO[3−2] detections with NOEMA. RO-1001 does not contain any detectable evidence of ongoing AGN activity, down to limits that are consistent with a cosmic average co-existence of an AGN and SFR at typical levels. The hosting DM halo mass is estimated to be typical for a group, with ∼4 × 10 13 M ⊙ consistently derived using three methods, including a blind X-ray measurement with significance of 3.1 σ . RO-1001 hosts a giant Ly α halo with a luminosity of 1.3 × 10 44 erg s −1 as
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