  We show that spin-orbit coupling (SOC) gives rise to pairing instability in a
highly polarized two-dimensional Fermi gas for arbitrary interaction strength.
The pairing instability can lead to a Fulde-Ferrell-Larkin-Ovchinnikov-like
molecular state, which undergoes a first-order transition into a pairing state
with zero center-of-mass momentum as the parameters are tuned. These pairing
states are metastable against a polaron state dressed by particle-hole
fluctuations for small SOC. At large SOC, a polaron-molecule transition exists,
which suggests a phase transition between the topological superfluid state and
the normal state for a highly polarized Fermi gas in the thermodynamic limit.
As polarization in a Fermi gas with SOC is induced by the effective Zeeman
field, we also discuss the influences of the effective Zeeman field on the
ground state of the system. Our findings may be tested directly in future
experiments.
