  To establish the connection between galaxies and UV-detected absorption
systems in the local universe, a deep ($g\leq20$) 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
$z\leq0.1$ 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 $L\geq0.3\,L^*$ 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.
