Formation of super-massive black holes in galactic nuclei I: delivering
  seed intermediate-mass black holes in massive stellar clusters
  Supermassive black holes (SMBHs) are found in most galactic nuclei. A
significant fraction of these nuclei also contain a nuclear stellar cluster
(NSC) surrounding the SMBH. In this paper, we consider the idea that the NSC
forms first, from the merger of several stellar clusters that may contain
intermediate-mass black holes (IMBHs). These IMBHs can subsequently grow in the
NSC and form an SMBH. We carry out $N$-body simulations of the simultaneous
merger of three stellar clusters to form an NSC, and investigate the outcome of
simulated runs containing zero, one, two and three IMBHs. We find that IMBHs
can efficiently sink to the centre of the merged cluster. If multiple merging
clusters contain an IMBH, we find that an IMBH binary is likely to form and
subsequently merge by gravitational wave emission. We show that these mergers
are catalyzed by dynamical interactions with surrounding stars, which
systematically harden the binary and increase its orbital eccentricity. The
seed SMBH will be ejected from the NSC by the recoil kick produced when two
IMBHs merge, if their mass ratio $q\gtrsim 0.15$. If the seed is ejected then
no SMBH will form in the NSC. This is a natural pathway to explain those
galactic nuclei that contain an NSC but apparently lack an SMBH, such as M33.
However, if an IMBH is retained then it can seed the growth of an SMBH through
gas accretion and tidal disruption of stars.
