  We present a detailed study of the population imbalanced three-component
Hubbard chain with attractive interactions. Such a system can be realized
experimentally with three different hyperfine states of ultra cold $^6$Li atoms
in an optical lattice. We find that there are different phases that compete
with each other in this system: A molecular superfluid phase in which the three
fermion species pair up to form molecules (trions), a usual pairing phase
involving two species with exactly opposite momenta, and a more exotic
generalized Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase consisting of three
competing pairing tendencies with different non-zero center-of-mass momenta. At
large attractive interactions the system exhibits strong tendencies towards
collapse and phase separation. Employing the
density-matrix-renormalization-group-method (DMRG) to determine the decay
exponents of the various correlators we establish the phase diagram of this
model for different fillings and interactions. We also discuss the
experimentally relevant situation in a trap and report the existence of an
additional region where two species are dynamically balanced.
