  The origin of the black hole (BH) binary mergers observed by LIGO-Virgo is
still uncertain, as are the boundaries of the stellar BH mass function. Stellar
evolution models predict a dearth of BHs both at masses $\gtrsim 50$ M$_\odot$
and $\lesssim 5$ M$_\odot$, thus leaving low- and high-mass gaps in the BH mass
function. A natural way to form BHs of these masses is through mergers of
neutron stars (NSs; for the low-mass gap) or lower-mass BHs (for the high-mass
gap); the low- or high-mass-gap BH produced as a merger product can then be
detected by LIGO-Virgo if it merges again with a new companion. We show that
the evolution of a 2+2 quadruple system can naturally lead to BH mergers with
component masses in the low- or high-mass gaps. In our scenario, the BH in the
mass gap originates from the merger of two NSs, or two BHs, in one of the two
binaries and the merger product is imparted a recoil kick (from anisotropic
gravitational wave emission), which triggers its interaction with the other
binary component of the quadruple system. The outcome of this three-body
interaction is usually a new eccentric compact binary containing the BH in the
mass gap, which can then merge again. The merger rate is $\sim 10^{-7} -
10^{-2}$ Gpc$^{-3}$ yr$^{-1}$ and $\sim 10^{-3} - 10^{-2}$ Gpc$^{-3}$ yr$^{-1}$
for BHs in the low-mass and high-mass gap, respectively. As the sensitivity of
gravitational wave detectors improves, tighter constraints will soon be placed
on the stellar BH mass function.
