The nature of compact groups (CGs) of galaxies, apparently so dense that the galaxies often overlap, is still a subject of debate: Are CGs roughly as dense in 3D as they appear in projection? Or are they caused by chance alignments of galaxies along the line-of-sight, within larger virialized groups or even longer filamentary structures? The nature of CGs is re-appraised using the z=0 outputs of three galaxy formation models, applied to the dissipationless Millennium Simulation. The same selection criteria are applied to mock galaxy catalogs from these models as have been applied by Hickson and co-workers in redshift space. We find 20 times as many mock CGs as the `HCGs found by Hickson within a distance corresponding to 9000 km/s. This very low (5%) HCG completeness is caused by Hickson missing groups that were either faint, near the surface brightness threshold, of small angular size, or with a dominant brightest galaxy. We find that most velocity-filtered CGs are physically dense, regardless of the precise threshold used in 3D group size and line-of-sight elongation, and of the galaxy formation model used. This result also holds for mock CGs with the same selection biases as was found for the HCGs.
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