We cross-correlate QSOs from the 2dF Survey with galaxy groups. The galaxy samples are limited to B < 20.5. We use an objective algorithm to detect galaxy groups. A 3sigma anti-correlation is observed between QSOs and galaxy groups. This paucity of faint QSOs around groups is neither a selection effect nor due to restrictions on the placement of 2dF fibres. By observing the colours of QSOs on the scales of the anti-correlation, we limit dust in galaxy groups, finding a maximum reddening of E(b_j-r) < 0.012 at the 95% level. The small amount of dust thus inferred is insufficient to cause the anti-correlation, supporting the suggestion by Croom & Shanks that the signal is due to gravitational lensing. The possibility remains that tailored dust models, such as grey dust, heavy patches of dust or a combination of dust and lensing, could explain the anti-correlation. Assuming the signal is caused by lensing rather than dust, we measure the average velocity dispersion of a Singular Isothermal Sphere that would cause the anti-correlation as around 1150 km/s. Simulations reject 600 km/s at the 5% significance level. We also model foreground lenses as NFW haloes and measure the typical mass within 1.5 Mpc/h of the halo centre as M_{1.5} = (1.2 +/- 0.9) x 10^{15} solarmasses/h. Regardless of whether we utilise a SIS or NFW dark matter profile, our model favours more mass in groups than accounted for in a universe with density parameter Omega_m = 0.3. Detailed simulations and galaxy group redshifts will significantly reduce the current systematic uncertainties in these $Omega_m$ estimates. Reducing the remaining uncertainty will require larger QSO and galaxy group surveys (abridged).
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