  Galaxies with stellar bulges are generically observed to host supermassive
black holes (SMBHs). The hierarchical merging of galaxies should therefore lead
to the formation of SMBH binaries. Merging of old massive galaxies with little
gas promotes the formation of low-density nuclei where SMBH binaries are
expected to survive over long times. If the binary lifetime exceeds the typical
time between mergers, then triple black hole (BH) systems may form. We study
the statistics of close triple-SMBH encounters in galactic nuclei by computing
a series of 3-body orbits with physically-motivated initial conditions
appropriate for giant elliptical galaxies. Our simulations include a smooth
background potential consisting of a stellar bulge plus a dark matter halo,
drag forces due to gravitational radiation and dynamical friction on the stars
and dark matter, and a simple model of the time evolution of the inner density
profile under heating and mass ejection by the SMBHs. We find that the binary
pair coalesces as a result of repeated close encounters in ~85% of our runs,
and in ~15% of cases a new eccentric binary forms from the third SMBH and
binary remnant and coalesces during the run time. In about 40% of the runs the
lightest BH is left wandering through the galactic halo or escapes the galaxy
altogether, but escape of all three SMBHs is exceedingly rare. The triple
systems typically scour out cores with mass deficits ~1-2 times their total
mass, which can help to account for the large cores observed in some massive
elliptical galaxies, such as M87. The high coalescence rate, prevalence of very
high-eccentricity orbits, and gravitational radiation ``spikes'' during close
encounters in our runs, may provide interesting signals for the future Laser
Interferometer Space Antenna (LISA).
