  Active galactic nuclei (AGN) probably control the growth of their host
galaxies via feedback in the form of wide-angle wind-driven outflows. These
establish the observed correlations between supermassive black hole (SMBH)
masses and host galaxy properties, e.g. the spheroid velocity dispersion
$\sigma$. In this paper we consider the growth of the SMBH once it starts
driving a large-scale outflow through the galaxy. To clear the gas and
ultimately terminate further growth of both the SMBH and the host galaxy, the
black hole must continue to grow its mass significantly, by up to a factor of a
few, after reaching this point. The mass increment $\Delta M_{\rm BH}$ depends
sensitively on both galaxy size and SMBH spin. The galaxy size dependence leads
to $\Delta M_{\rm BH} \propto \sigma^5$ and a steepening of the $M-\sigma$
relation beyond the analytically calculated $M \propto \sigma^4$, in agreement
with observation. Slowly--spinning black holes are much less efficient in
producing feedback, so at any given $\sigma$ the slowest--spinning black holes
should be the most massive. Current observational constraints are consistent
with this picture, but insufficient to test it properly; however, this should
change with upcoming surveys.
