Some previous investigations have found that the fraction (f_AGN) of active galactic nuclei (AGNs) is lower in clusters than in the field. This can result from the suppression of galaxy-galaxy mergers in high-velocity dispersion (sigma_v) clusters, if the formation and/or fueling of AGNs is directly related to the merging process. We investigate the existence of a relation between f_AGN and sigma_v in galaxy clusters in order to shed light on the formation and evolution processes of AGNs and cluster galaxies. Using data from the Sloan Digital Sky Survey we determine f_AGN and sigma_v for the clusters in two samples, extracted from the catalogs of Popesso et al. (2006a) and Miller et al. (2005), and excluding clusters with significant evidence for substructures. We find a significant f_AGN-sigma_v anti-correlation. Clusters with sigma_v lower and, respectively, higher than 500 km/s have AGN fractions of $0.21 pm 0.01$ and $0.15 pm 0.01$, on average. The f_AGN-sigma_v relation can be described by a model that assumes f_AGN is proportional to the galaxies merging rate, plus a constant. Since f_AGN increases with decreasing sigma_v, AGNs are likely to have played a significant role in heating the intra-cluster medium and driving galaxy evolution in cluster precursors and groups.
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