To better understand the mechanism or mechanisms that lead to AGN activity today, we measure the X-ray AGN fraction in a new sample of nearby clusters and examine how it varies with galaxy properties, projected cluster-centric radius, and cluster velocity dispersion. We present new wide-field Chandra X-ray Observatory observations of Abell 85, Abell 754 and the background cluster Abell 89B out to their virial radii. Out of seventeen X-ray sources associated with galaxies in these clusters, we classify seven as X-ray AGN with L_{X,B} > 10^{41} erg/s. Only two of these would be classified as AGN based on their optical spectra. We combine these observations with archival data to create a sample of X-ray AGN from six z < 0.08 clusters and find that 3.4+1.1/-0.8% of M_R < -20 galaxies host X-ray AGN with L_{X,B} > 10^{41} erg/s. We find that more X-ray AGN are detected in more luminous galaxies and attribute this to larger spheriods in more luminous galaxies and increased sensitivity to lower Eddington-rate accretion from black holes in those spheroids. At a given X-ray luminosity limit, more massive black holes can be accreting less efficiently, yet still be detected. If interactions between galaxies are the principal drivers of AGN activity, then the AGN fraction should be higher in lower velocity dispersion clusters and the outskirts of clusters. However, the tendency of the most massive and early-type galaxies to lie in the centers of the richest clusters could dilute such trends. While we find no variation in the AGN fraction with projected cluster-centric radius, we do find that the AGN fraction increases significantly from 2.6+1.0/-0.8% in rich clusters to 10.0+6.2/-4.3% in those with lower velocity dispersions.
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