The M - \sigma relation in different environments
  Galaxies become red and dead when the central supermassive black hole (SMBH)
becomes massive enough to drive an outflow beyond the virial radius of the
halo. We show that this final SMBH mass is larger than the final SMBH mass in
the bulge of a spiral galaxy by up to an order of magnitude. The M - \sigma
relations in the two galaxy types are almost parallel (M \propto
\sigma^{4+\beta}, with \beta < 1) but offset in normalization, with the extra
SMBH mass supplied by the major merger transforming the galaxy into an
elliptical, or by mass gain in a galaxy cluster. This agrees with recent
findings that SMBH in two Brightest Cluster Galaxies are \sim 10\times the
expected M-\sigma mass. We show that these results do not strongly depend on
the assumed profile of the dark matter halo, so analytic estimates found for an
isothermal potential are approximately valid in all realistic cases.
  Our results imply that there are in practice actually {\it three} M - \sigma
relations, corresponding to spiral galaxies with evolved bulges, field
elliptical galaxies and cluster centre elliptical galaxies. A fourth relation,
corresponding to cluster spiral galaxies, is also possible, but such galaxies
are expected to be rare. All these relations have the form M_{\rm BH} =
C_n\sigma^4, with only slight difference in slope between field and cluster
galaxies, but with slightly different coefficients C_n. Conflating data from
galaxies of different types and fitting a single relation to them tends to
produce a higher power of \sigma.
