We compare accretion and black hole spin as potential energy sources for outbursts from AGN in brightest cluster galaxies (BCGs). Based on our adopted spin model, we find that the distribution of AGN power estimated from X-ray cavities is consistent with a broad range of both spin parameter and accretion rate. Sufficient quantities of molecular gas are available in most BCGs to power their AGN by accretion alone. However, we find no correlation between AGN power and molecular gas mass over the range of jet power considered here. For a given AGN power, the BCGs gas mass and accretion efficiency, defined as the fraction of the available cold molecular gas that is required to power the AGN, both vary by more than two orders of magnitude. Most of the molecular gas in BCGs is apparently consumed by star formation or is driven out of the nucleus by the AGN before it reaches the nuclear black hole. Bondi accretion from hot atmospheres is generally unable to fuel powerful AGN, unless their black holes are more massive than their bulge luminosities imply. We identify several powerful AGN that reside in relatively gas-poor galaxies, indicating an unusually efficient mode of accretion, or that their AGN are powered by another mechanism. If these systems are powered primarily by black hole spin, rather than by accretion, spin must also be tapped efficiently in some systems, i.e., $P_{rm jet} > dot Mc^2$, or their black hole masses must be substantially larger than the values implied by their bulge luminosities. We constrain the (model dependent) accretion rate at the transition from radiatively inefficient to radiatively efficient accretion flows to be a few percent of the Eddington rate, a value that is consistent with other estimates.
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