  Recent detections of gravitational waves from mergers of neutron stars (NSs)
and black holes (BHs) in the low and high-end mass gap regimes (with masses
between ~2-5 M_Sun and exceeding ~50 M_Sun, respectively) pose a puzzle to
standard stellar and binary evolution theory. Mass-gap mergers may originate
from successive mergers in hierarchical systems such as quadruples. Here, we
consider repeated mergers of NSs and BHs in stellar 2+2 quadruple systems.
Under certain circumstances, secular evolution acts to accelerate the merger of
one of the inner binaries. Subsequently, the merger remnant may interact with
the companion binary, yielding a second-generation merger event. We model the
initial stellar and binary evolution of the inner binaries as isolated systems.
In the case of successful compact object formation, we subsequently follow the
secular dynamical evolution of the quadruple system. When a merger occurs, we
take into account merger recoil, and model subsequent evolution using direct
N-body methods. With different assumptions on the initial binary properties, we
find that the majority of first-generation mergers are not much affected by
secular evolution, with their observational properties mostly consistent with
isolated binaries. A small subset shows imprints of secular evolution through
residual eccentricity in the LIGO band, and retrograde spin-orbit orientations.
Second-generation mergers can be strongly affected by scattering (i.e.,
three-body interactions) induced by the first-generation merger. In particular,
scattering can account for mergers within the low-end mass gap, although not
the high-end mass gap. Also, scattering could explain highly eccentric LIGO
sources and negative effective spin parameters.
