Context : 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 .

Aims : 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 .

Methods : 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 .

Results : We find a significant f _ { AGN } - \sigma _ { v } anti-correlation .
Clusters with \sigma _ { v } lower and , respectively , higher than 500 km s ^ { -1 } 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 .

Conclusions : Since f _ { AGN } increases with decreasing \sigma _ { v } , AGNs are likely to have played a significant rôle in heating the intra-cluster medium and driving galaxy evolution in cluster precursors and groups .