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We present an analysis of the growth of black holes through accretion andbulges through star formation in 33 galaxies at the centers of cooling flows.Most of these systems show evidence of cavities in the intracluster medium(ICM) inflated by radio jets emanating from their active galactic nuclei (AGN).We present a new and extensive analysis of X-ray cavities in these systems. Wefind that AGN are energetically able to balance radiative losses (cooling) fromthe ICM in more than half of our sample. Using a subsample of 17 systems, weexamine the relationship between cooling and star formation. We find that thestar formation rates are approaching or are comparable to X-ray and far UVlimits on the rates of gas condensation onto the central galaxy. The remainingradiative losses could be offset by AGN feedback. The vast gulf betweenradiative losses and the sink of cooling material, which has been the primaryobjection to cooling flows, has narrowed and, in some cases, is no longer aserious issue. Using the cavity (jet) powers, we place strong lower limits onthe rate of growth of supermassive black holes in central galaxies, and we findthat they are growing at an average rate of ~ 0.1 solar masses per year, withsome systems growing as quickly as ~ 1 solar mass per year. We find a trendbetween bulge growth (star formation) and black hole growth that isapproximately in accordance with the slope of the local (Magorrian) relationbetween black hole and bulge mass. However, the large scatter in the trendsuggests that bulges and black holes do not always grow in lock step.(Abridged)Comment: 17 pages, 6 figures, accepted to ApJ. Minor changes to text and  figure

The Feedback‐regulated Growth of Black Holes and Bulges through Gas Accretion and Starbursts in Cluster Central Dominant Galaxies