The phase transition between galaxies and quasars is often identified with the rare population of hyper-luminous, hot dust-obscured galaxies. Galaxy formation models predict these systems to grow via mergers, that can deliver large amounts of gas toward their centers, induce intense bursts of star formation and feed their supermassive black holes. Here we report the detection of 24 galaxies emitting Lyman-α emission on projected physical scales of about 400 kpc around the hyper-luminous hot dust-obscured galaxy W0410-0913, at redshift z = 3.631, using Very Large Telescope observations. While this indicates that W0410-0913 evolves in a very dense environment, we do not find clear signs of mergers that could sustain its growth. Data suggest that if mergers occurred, as models expect, these would involve less massive satellites, with only a moderate impact on the internal interstellar medium of W0410-0913, which is sustained by a rotationally-supported fast-rotating molecular disk, as Atacama Large Millimeter Array observations suggest.