  Nuclear star clusters that surround supermassive black holes (SMBHs) in
galactic nuclei are thought to contain large numbers of black holes (BHs) and
neutron stars (NSs), a fraction of which form binaries and could merge by
Kozai-Lidov oscillations (KL). Triple compact objects are likely to be present,
given what is known about the multiplicity of massive stars, whose life ends
either as a NS or a BH. In this paper, we present a new possible scenario for
merging BHs and NSs in galactic nuclei. We study the evolution of a triple
black hole (BH) or neutron star (NS) system orbiting an SMBH in a galactic
nucleus by means of direct high-precision $N$-body simulations, including
Post-Newtonian terms. We find that the four-body dynamical interactions can
increase the KL angle window for mergers compared to the binary case and make
BH and NS binaries merge on shorter timescales. We show that the merger
fraction can be up to $\sim 5$--$8$ times higher for triples than for binaries.
Therefore, even if the triple fraction is only $\sim 10\%$--$20\%$ of the
binary fraction, they could contribute to the merger events observed by
LIGO/VIRGO in comparable numbers.
