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We have conducted a long slit search for low surface brightness Lyman-alphaemitters at redshift 2.67 < z < 3.75. A 92 hour long exposure with VLT/FORS2down to a 1-sigma surface brightness detection limit of 8x10^-20erg/cm2/s/sqarcsec yielded a sample of 27 single line emitters with fluxes of afew times 10^-18 erg/s/cm2. We present arguments that most objects are indeedLyman-alpha. The large comoving number density, the large covering factor,dN/dz ~ 0.2-1, and the often extended Lyman-alpha emission suggest that theemitters be identified with the elusive host population of damped Lyman-alphasystems (DLAS) and high column density Lyman limit systems. A small inferredstar formation rate, perhaps supplanted by cooling radiation, appears toenergetically dominate the Lyman-alpha emission, and is consistent with the lowmetallicity, low dust content, and theoretically inferred low masses of DLAS,and with the relative lack of success of earlier searches for their opticalcounterparts. (abridged)

A Population of Faint Extended Line Emitters and the Host Galaxies of Optically Thick QSO Absorption Systems