We exploit ALMA 870um observations to measure the star-formation rates (SFRs) of eight X-ray detected Active Galactic Nuclei (AGNs) in a z~3.1 protocluster, four of which reside in extended Ly-alpha haloes (often termed Ly-alpha blobs: LABs). Three of the AGNs are detected by ALMA and have implied SFRs of ~220-410~M_sun/yr; the non detection of the other five AGNs places SFR upper limits of <210 M_sun/yr. The mean SFR of the protocluster AGNs (~110-210 M_sun/yr) is consistent (within a factor of ~0.7-2.3) with that found for co-eval AGNs in the field, implying that galaxy growth is not significantly accelerated in these systems. However, when also considering ALMA data from the literature, we find evidence for elevated mean SFRs (up-to a factor of ~5.9 over the field) for AGNs at the protocluster core, indicating that galaxy growth is significantly accelerated in the central regions of the protocluster. We also show that all of the four protocluster LABs are associated with an ALMA counterpart within the extent of their Ly-alpha emission. The SFRs of the ALMA sources within the LABs (~150-410 M_sun/yr) are consistent with those expected for co-eval massive star-forming galaxies in the field. Furthermore, the two giant LABs (with physical extents of >100 kpc) do not host more luminous star formation than the smaller LABs, despite being an order of magnitude brighter in Ly-alpha emission. We use these results to discuss star formation as the power source of LABs.
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