A small survey of the UV-absorbing gas in 12 low-$z$ galaxy groups has been conducted using the Cosmic Origins Spectrograph (COS) on-board the Hubble Space Telescope (HST). Targets were selected from a large, homogeneously-selected sample of groups found in the Sloan Digital Sky Survey (SDSS). A critical selection criterion excluded sight lines that pass close ($<1.5$ virial radii) to a group galaxy, to ensure absorber association with the group as a whole. Deeper galaxy redshift observations are used both to search for closer galaxies and also to characterize these $10^{13.5}$ to $10^{14.5} M_{odot}$ groups, the most massive of which are highly-virialized with numerous early-type galaxies (ETGs). This sample also includes two spiral-rich groups, not yet fully-virialized. At group-centric impact parameters of 0.3-2 Mpc, these $mathrm{S/N}=15$-30 spectra detected HI absorption in 7 of 12 groups; high (OVI) and low (SiIII) ion metal lines are present in 2/3 of the absorption components. None of the three most highly-virialized, ETG-dominated groups are detected in absorption. Covering fractions $gtrsim50$% are seen at all impact parameters probed, but do not require large filling factors despite an enormous extent. Unlike halo clouds in individual galaxies, group absorbers have radial velocities which are too low to escape the group potential well without doubt. This suggests that these groups are closed boxes for galactic evolution in the current epoch. Evidence is presented that the cool and warm group absorbers are not a pervasive intra-group medium (IGrM), requiring a hotter ($Tsim10^6$ to $10^7$ K) IGrM to be present to close the baryon accounting.
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