  Recent observations support the idea that nuclear black holes grew by gas
accretion while shining as luminous quasars at high redshift, and they
establish a relation of the black hole mass with the host galaxy's spheroidal
stellar system. We develop an analytic model to calculate the expected impact
of mergers on the masses of black holes in massive clusters of galaxies. We use
the extended Press-Schechter formalism to generate Monte Carlo merger histories
of halos with a mass 10^{15} h^{-1} Msun. We assume that the black hole mass
function at z=2 is similar to that inferred from observations at z=0 (since
quasar activity declines markedly at z<2), and we assign black holes to the
progenitor halos assuming a monotonic relation between halo mass and black hole
mass. We follow the dynamical evolution of subhalos within larger halos,
allowing for tidal stripping, the loss of orbital energy by dynamical friction,
and random orbital perturbations in gravitational encounters with subhalos, and
we assume that mergers of subhalos are followed by mergers of their central
black holes. Our analytic model reproduces numerical estimates of the subhalo
mass function. We find that the most massive black holes in massive clusters
typically grow by a factor ~ 2 by mergers after gas accretion has stopped. In
our ten realizations of 10^{15} h^{-1} Msun clusters, the highest initial (z=2)
black hole masses are 5-7 x 10^9 Msun, but four of the clusters contain black
holes in the range 1-1.5 x 10^{10} Msun at z=0. Satellite galaxies may host
black holes whose mass is comparable to, or even greater than, that of the
central galaxy. Thus, black hole mergers can significantly extend the very high
end of the black hole mass function.
