[Abridged] We present K-band data for the brightest cluster galaxies (BCGs) from the ESO Distant Cluster Survey. These data are combined with photometry from Aragon-Salamanca et al. (1998) and a low-redshift comparison sample from von der Linden et al. (2007). The K-band Hubble diagram for BCGs exhibits very low scatter (~0.35mag) since z=1. The colour and $K$-band luminosity evolution of the BCGs are in good agreement with passively-evolving stellar populations formed at z>2. We do not detect any significant change in the stellar mass of the BCG since z~1. These results do not seem to depend on the velocity dispersion of the parent cluster. There is a correlation between the 1D velocity dispersion of the clusters and the K-band luminosity of the BCGs (after correcting for passive evolution). The clusters with large velocity dispersions tend to have brighter BCGs, i.e., BCGs with larger stellar masses. This dependency, although significant, is relatively weak: the stellar mass of the BCGs changes only by ~70% over a two-order-of-magnitude range in cluster mass. This dependency doesnt change significantly with redshift. The models of De Lucia & Blaizot (2007) predict colours which are in reasonable agreement with the observations because the growth in stellar mass is dominated by the accretion of old stars. However, the stellar mass in the model BCGs grows by a factor of 3-4 since z=1, a growth rate which seems to be ruled out by the observations. The models predict a dependency between the BCGs stellar mass and the velocity dispersion of the parent cluster in the same sense as the data, but the dependency is significantly stronger than observed. However, one major difficulty in this comparison is that we have measured fixed metric aperture magnitudes while the models compute total luminosities.
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