  Merging compact black-hole (BH) binaries are likely to exist in the nuclear
star clusters around supermassive BHs (SMBHs), such as Sgr A$^\ast$. They may
also form in the accretion disks of active galactic nuclei. Such compact
binaries can emit gravitational waves (GWs) in the low-frequency band (0.001-1
Hz) that are detectable by several planned space-borne GW observatories. We
show that the orbital axis of the compact binary may experience significant
variation due to the frame-dragging effect associated with the spin of the
SMBH. The dynamical behavior of the orbital axis can be understood analytically
as a resonance phenomenon. We show that rate of change of the binary orbital
axis encodes the information on the spin of the SMBH. Therefore detecting GWs
from compact binaries around SMBHs, particularly the modulation of the waveform
associated with the variation of the binary orbital axis, can provide a new
probe on the spins of SMBHs.
