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We investigate the evolution of intermediate-mass (IMBHs), stellar (BHs) andbinary black holes (BHBs), deposited near a supermassive black hole (SMBH) by apopulation of massive star clusters. Stellar BHs rapidly segregate around theSMBH, driving the formation of extreme mass-ratio inspirals that coalesce at arate $\Gamma= 0.02-0.2$ yr$^{-1}$ Gpc$^{-3}$ at redshift $z=0$. A few IMBHsorbiting the SMBH favour the formation of massive pairs that coalescence withina Hubble time, being the merger rate for this channel $\Gamma =0.03$ yr$^{-1}$Gpc$^{-3}$. Recoiling kicks post-merger can eject the remnant from the galaxycentre, especially in dwarf galaxies. Our results suggest that this mechanismcan lead to up to $10^5$ ejected SMBH within 1 Gpc. An IMBH co-existing with afew single and binary BHs in the same cluster can affect significantly theirevolution, either driving binary disruption, yielding to intermediate-massratio inspirals (merger rate $\Gamma =9.5$ yr$^{-1}$ Gpc$^{-3}$), or boostingBHBs coalescence ($\Gamma =2-8$ yr$^{-1}$ Gpc$^{-3}$). In a few simulations,the SMBH boosts BHBs coalescence, leading this process to a merger rate $\Gamma=1$ yr$^{-1}$ Gpc$^{-3}$. We note that BHBs experiencing a merger in a galacticnucleus can be erroneously estimated $\sim 30\%$ heavier than it really isbecause of the Doppler shift of the wave frequency as caused by the rapidmotion around the SMBH. All our simulations are carried out using an $N$-bodycode tailored to treat close encounters and post-Newtonian dynamics, thatincludes also the galaxy field and dynamical friction in the particles'equation of motion.

The MEGaN project II. Gravitational waves from intermediate mass- and binary black holes around a supermassive black hole