  The astrophysical origin of gravitational wave (GW) events discovered by
LIGO/VIRGO remains an outstanding puzzle. In active galactic nuclei (AGN),
compact-object binaries form, evolve, and interact with a dense star cluster
and a gas disk. An important question is whether and how binaries merge in
these environments. To address this question, we have performed one-dimensional
$N$-body simulations combined with a semi-analytical model which includes the
formation, disruption, and evolution of binaries self-consistently. We point
out that binaries can form in single-single interactions by the dissipation of
kinetic energy in a gaseous medium. This ``gas capture'' binary formation
channel contributes up to $97\,\%$ of gas-driven mergers and leads to a high
merger rate in AGN disks even without pre-existing binaries. We find the merger
rate to be in the range $\sim 0.02-60\,\mathrm{Gpc^{-3}yr^{-1}}$. The results
are insensitive to the assumptions on gaseous hardening processes: we find that
once they are formed, binaries merge efficiently via binary-single interactions
even if these gaseous processes are neglected. We find that the average number
of mergers per BH is $0.4$, and the probability for repeated mergers in 30 Myr
is $\sim 0.21-0.45$. High BH masses due to repeated mergers, high
eccentricities, and a significant Doppler drift of GWs are promising signatures
which distinguish this merger channel from others. Furthermore, we find that
gas-capture binaries reproduce the distribution of LMXBs in the Galactic
center, including an outer cutoff at $\sim1$ pc due to the competition between
migration and hardening by gas torques.
