POSSIBLE IDENTIFICATION OF A CLUSTER OF GALAXIES AT REDSHIFT Z = 3.4

We report the possible detection of a cluster of 16 radio-quiet galaxi es at z = 3.4, identified in the field around the optically thick abso rption system toward Q0000 - 263 at z(abs) = 3.390. Two of them, a Lya lpha emitter at z = 3.428 and the galaxy responsible for the absorptio n system, have redshifts spectroscopically confirmed. The other 14 gal axies identified using a multicolor imaging technique designed to dete ct sources in the redshift interval 3 approximately-less-than z approx imately-less-than 3.5 which are characterized by a Lyman discontinuity in an otherwise flat spectrum, have broad-band spectral energy distri butions identical to the two galaxies with known redshift. They are sp atially distributed in two apparent clumps, around the damped absorber and the Lyalpha galaxy, respectively. A clustering analysis excludes with 98.8% confidence that this association is a realization of a Pois sonian distribution and confirms that the observed clumps are real. Th e implications are that the 16 galaxies are members of a cluster at z almost-equal-to 3.4, by far the most distant ever detected. An estimat e of the mass bounded in stars of this cluster is 3 x 10(12) M. (q0 = 0 and H0 = 50 km s-1 Mpc-1 throughout this Letter), while the total ma ss (baryonic + dark) is 6 x 10(14) M.. We also estimate that at z = 3. 4 the correlation length is 2.2 Mpc, which, compared to the present va lue of 11 Mpc, suggests that the clustering evolution is still close t o the linear regime.