  Galactic nuclei are promising sites for stellar origin black hole (BH)
mergers, as part of merger hierarchies in deep potential wells. We show that
binary black hole (BBH) merger rates in active galactic nuclei (AGN) should
always exceed merger rates in quiescent galactic nuclei (nuclear star clusters,
NSCs) around supermassive BHs (SMBHs) without accretion disks. This is
primarily due to average binary lifetimes in AGN that are significantly shorter
than in NSCs. The lifetime difference comes from rapid hardening of BBHs in
AGN, such that their semi-major axes are smaller than the hard-soft boundary of
their parent NSC; this contrasts with the large average lifetime to merger for
BBHs in NSCs around SMBHs, due to binary ionization mechanisms. Secondarily,
merger rates in AGNs are enhanced by gas-driven binary formation mechanisms.
Formation of new BHs in AGN disks are a minor contributor to the rate
differences. With the gravitational wave detection of several BBHs with at
least one progenitor in the upper mass gap, and signatures of dynamical
formation channels in the $\chi_{\rm eff}$ distribution, we argue that AGN
could contribute $\sim 25\%-80\%$ of the LIGO-Virgo measured rate of $\sim 24
\rm{Gpc}^{-3} \rm{yr}^{-1}$.
