We have observed 433 z<=0.08 brightest cluster galaxies (BCGs) in a full-sky survey of Abell clusters. The BCG Hubble diagram is consistent to within 2% of a Omega_m=0.3, Lambda=0.7 Hubble relation. The L_m-alpha relation for BCGs, which uses alpha, the log-slope of the BCG photometric curve of growth, to predict metric luminosity, L_m, has 0.27 mag residuals. We measure central stellar velocity dispersions, sigma, of the BCGs, finding the Faber-Jackson relation to flatten as the metric aperture grows to include an increasing fraction of the total BCG luminosity. A 3-parameter metric plane relation using alpha and sigma together gives the best prediction of L_m, with 0.21 mag residuals. The projected spatial offset, r_x, of BCGs from the X-ray-defined cluster center is a gamma=-2.33 power-law over 1<r_x<10^3 kpc. The median offset is ~10 kpc, but ~15% of the BCGs have r_x>100 kpc. The absolute cluster-dispersion normalized BCG peculiar velocity |Delta V_1|/sigma_c follows an exponential distribution with scale length 0.39+/-0.03. Both L_m and alpha increase with sigma_c. The alpha parameter is further moderated by both the spatial and velocity offset from the cluster center, with larger alpha correlated with the proximity of the BCG to the cluster mean velocity or potential center. At the same time, position in the cluster has little effect on L_m. The luminosity difference between the BCG and second-ranked galaxy, M2, increases as the peculiar velocity of the BCG within the cluster decreases. Further, when M2 is a close luminosity rival of the BCG, the galaxy that is closest to either the velocity or X-ray center of the cluster is most likely to have the larger alpha. We conclude that the inner portions of the BCGs are formed outside the cluster, but interactions in the heart of the galaxy cluster grow and extend the envelopes of the BCGs.
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