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 dis-sipationless 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~(-1). 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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