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We report the likely detection of a cluster of 16 radio-quiet galaxies at
redshift $z=3.4$, identified in the field around the optically thick absorption
system toward Q0000-263 at $z_{abs}=3.390$. Two of these galaxies, a Ly$\alpha$
emitter at $z_{em}=3.428$ and the one responsible for the absorption system,
have redshifts spectroscopically confirmed. The other 14 galaxies have been
identified with a multi--color imaging technique that makes use of passbands
($U_n$, $G$ and ${\cal R}$ properly designed to detect sources in the redshift
interval $3
In addition, for the first time, a significant number of high-redshift
radio-quiet galaxies is available for establishing the general properties
of young, normal galaxies. Their rest-frame UV SED is very blue, basically
consistent with a flat spectrum (after accounting for Lyman $\alpha$ forest
blanketing), which is typical of star--forming galaxies. The SFRs measured
from the continuum luminosity range between 30 and 100 M$_{\odot}$y$^{-1}$.