[Abridged] We characterise a massive proto-cluster at z=2.895 that we found in the COSMOS field using the spectroscopic sample of the VIMOS Ultra-Deep Survey (VUDS). This is one of the rare structures at z~3 not identified around AGNs or radio galaxies, so it is an ideal laboratory to study galaxy formation in dense environments. The structure comprises 12 galaxies with secure spectroscopic redshift in an area of 7x8, in a z bin of Dz=0.016. The measured galaxy number overdensity is delta_g=12+/-2. This overdensity has total mass of M~8.1x10^(14)M_sun in a volume of 13x15x17 Mpc^3. Simulations indicate that such an overdensity at z~2.9 is a proto-cluster that will collapse in a cluster of total mass M~2.5x10^(15)M_sun at z=0. We compare the properties of the galaxies within the overdensity with a control sample at the same z but outside the overdensity. We did not find any statistically significant difference between the properties (stellar mass, SFR, sSFR, NUV-r, r-K) of the galaxies inside and outside the overdensity. The stacked spectrum of galaxies in the overdensity background shows a significant absorption feature at the wavelength of Lya redshifted at z=2.895 (lambda=4736 A), with a rest frame EW = 4+/- 1.4 A. Stacking only background galaxies without intervening sources at z~2.9 along their line of sight, we find that this absorption feature has a rest frame EW of 10.8+/-3.7 A, with a detection S/N of ~4. These EW values imply a high column density (N(HI)~3-20x10^(19)cm^(-2)), consistent with a scenario where such absorption is due to intervening cold gas streams, falling into the halo potential wells of the proto-cluster galaxies. However, we cannot exclude the hypothesis that this absorption is due to the diffuse gas within the overdensity.
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