  The isolated binary evolution model for merging neutron stars (NSs) involves
processes such as mass transfer, common-envelope evolution, and natal kicks,
all of which are poorly understood. Also, the predicted NS-NS merger rates are
typically lower than the rates inferred from the LIGO GW170817 event. Here, we
investigate merger rates of NS and black hole (BH)-NS binaries in hierarchical
triple-star systems. In such systems, the tertiary can induce Lidov-Kozai (LK)
oscillations in the inner binary, accelerating its coalescence, and potentially
enhancing compact object merger rates. However, since compact objects originate
from massive stars, the prior evolution should also be taken into account.
Natal kicks, in particular, could significantly reduce the rates by unbinding
the tertiary before it can affect the inner binary through LK evolution. We
carry out simulations of massive triples taking into account stellar evolution
starting from the main sequence, secular and tidal evolution, and the effects
of supernovae. For large NS birth kicks (\sigma_k = 265 km/s), we find that the
triple NS-NS merger rate (several hundred Gpc^{-3} yr^{-1}) is lower by a
factor of ~2-3 than the binary rate, but for no kicks (\sigma_k = 0 km/s), the
triple rate (several thousand Gpc^{-3} yr^{-1}) is comparable to the binary
rate. Our results indicate that a significant fraction of NS-NS mergers could
originate from triples if a substantial portion of the NS population is born
with low kick velocities, as indicated by other work. However, uncertainties
and open questions remain because of our simplifying assumption of dynamical
decoupling after inner binary interaction has been triggered.
