  We discuss the rich phase diagram of doped AB2 $t-J$ chains by using data
from density matrix renormalization group and exact diagonalization techniques.
The $J$ vs $\delta$ (hole doping) phase diagram exhibits regions of itinerant
ferrimagnetism, incommensurate, resonating valence bond and Nagaoka states,
phase separation, and Luttinger liquid (LL) physics. Several features are
highlighted, such as the modulated ferrimagnetic structure, the occurrence of
Nagaoka spin polarons in the underdoped regime and small values of $J=4t^2/U$,
where $t$ is the first-neighbor hopping amplitude and $U$ is the on-site
repulsive Coulomb interaction, incommensurate structures with nonzero
magnetization, and strong-coupling LL physics in the high-doped regime. We also
verify that relevant findings are in agreement with the corresponding findings
in square and n-leg ladder lattices. In particular, we mention the instability
of Nagaoka ferromagnetism against $J$ and $\delta$.
