To establish the connection between galaxies and UV-detected absorption systems in the local universe, a deep ($gleq20$) and wide ($sim20^{prime}$ radius) galaxy redshift survey is presented around 47 sight lines to UV-bright AGN observed by the Cosmic Origins Spectrograph (COS). Specific COS science team papers have used this survey to connect absorbers to galaxies, groups of galaxies, and large-scale structures, including voids. Here we present the technical details of the survey and the basic measurements required for its use, including redshifts for individual galaxies and uncertainties determined collectively by spectral class (emission-line, absorption-line, and composite spectra) and completeness for each sight line as a function of impact parameter and magnitude. For most of these sight lines the design criteria of $>90$% completeness over a $>1$ Mpc region down to $lesssim0.1,L^*$ luminosities at $zleq0.1$ allows a plausible association between low-$z$ absorbers and individual galaxies. Ly$alpha$ covering fractions are computed to approximate the star-forming and passive galaxy populations using the spectral classes above. In agreement with previous results, the covering fraction of star-forming galaxies with $Lgeq0.3,L^*$ is consistent with unity inside one virial radius and declines slowly to $>50$% at 4 virial radii. On the other hand, passive galaxies have lower covering fractions ($sim60$%) and a shallower decline with impact parameter, suggesting that their gaseous halos are patchy but have a larger scale-length than star-forming galaxies. All spectra obtained by this project are made available electronically for individual measurement and use.
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