The Lya line in the UV and the [CII] line in the FIR are widely used tools to identify galaxies and to obtain insights into ISM properties in the early Universe. By combining data obtained with ALMA in band 7 at ~ 320 GHz as part of the ALMA Large Program to INvestigate [CII] at Early Times (ALPINE) with spectroscopic data from DEIMOS at Keck, VIMOS and FORS2 at the VLT, we assembled a unique sample of 53 main-sequence star-forming galaxies at 4.4 < z < 6 in which we detect both the Lya line and the [CII]. We used [CII], observed with ALMA, as a tracer of the systemic velocity of the galaxies, and we find that 90% of the selected objects have Lya-[CII] velocity offsets in the range 0 < Dv_Lya-[CII] < 400 km/s, in line with the few measurements available so far in the early Universe, and significantly smaller than those observed at lower z. We observe ISM-[CII] offsets in the range -500 < Dv_ISM-[CII] < 0 km/s, in line with values at all redshifts. We find significant anticorrelations between Dv_Lya-[CII] and the Lya rest-frame equivalent width EW0(Lya) (or equivalently, the Lya escape fraction f_esc(Lya)). According to available models for the radiative transfer of Lya photons, the escape of Lya photons would be favored in galaxies with high outflow velocities, in agreement with our observations. The uniform shell model would also predict that the Lya escape in galaxies with slow outflows (0 < v_out < 300 km/s) is mainly determined by the neutral hydrogen column density (NHI), while the alternative model by Steidel+10 would favor a combination of NHI and covering fraction as driver of the Lya escape. We suggest that the observed increase in Lya escape that is observed between z~2 and z~6 is not due to a higher incidence of fast outflows at high redshift, but rather to a decrease in average NHI along the line of sight, or alternatively, a decrease in HI covering fraction. [abridged]
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