High-redshift, dust-obscured galaxies -- selected to be luminous in the radio but relatively faint at 850um -- appear to represent a different population from the ultra-luminous submillimeter- (submm-) bright population. They may be star-forming galaxies with hotter dust temperatures or they may have lower far-infrared luminosities and larger contributions from obscured active galactic nuclei (AGN). Here we present observations of three z~2 examples of this population, which we term submm-faint radio galaxies (SFRGs) in CO(3-2) using the IRAM Plateau de Bure Interferometer to study their gas and dynamical properties. We estimate the molecular gas mass in each of the three SFRGs (8.3x10^{9} M_odot, <5.6x10^{9} M_odot and 15.4x10^{9} M_odot, respectively) and, in the case of RG163655, a dynamical mass by measurement of the width of the CO(3-2) line (8x10^{10} csc^2i M_odot). While these gas masses are substantial, on average they are 4x lower than submm-selected galaxies (SMGs). Radio-inferred star formation rates (<SFR_radio>=970 M_odotyr) suggest much higher star-formation efficiencies than are found for SMGs, and shorter gas depletion time scales (~11 Myr), much shorter than the time required to form their current stellar masses (~160 Myr; ~10^{11} M_odot). By contrast, SFRs may be overestimated by factors of a few, bringing the efficiencies in line with those typically measured for other ultraluminous star-forming galaxies and suggesting SFRGs are more like ultraviolet- (UV-)selected star-forming galaxies with enhanced radio emission. A tentative detection of rga at 350um suggests hotter dust temperatures -- and thus similar gas-to-dust mass fractions -- as the SMGs. We conclude that SFRGs radio luminosities are larger than would naturally scale from local ULIRGs given their gas masses or gas fractions.
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