We show that the well-known translation invariant ground states and the recently
discovered kink and antikink ground states are the complete set of pure infinite-volume
ground states (in the sense of local stability) of the spin-S ferromagnetic XXZ chains with
Hamiltonian H=-sum_x [ S^1_x S^1_{x+1} + S^2_x S^2_{x+1} + Delta S^3_x S^3_{x+1} ], for all
Delta >1, and all S=1/2,1,3/2,.... For the isotropic model (Delta =1) we show that all
ground states are translation invariant. For the proof of these statements we propose a
strategy for demonstrating completeness of the list of the pure infinite-volume ground
states of a quantum many-body system, of which the present results for the XXX and XXZ
chains can be seen as an example. The result for Delta>1 can also be proved by an easy
extension to general $S$ of the method used in [T. Matsui, Lett. Math. Phys. 37 (1996) 397]
for the spin-1/2 ferromagnetic XXZ chain with $\Delta>1$. However, our proof is
different and does not rely on the existence of a spectral gap. In particular, it also
works to prove absence of non-translationally invariant ground states for the isotropic
chains (Delta=1), which have a gapless excitation spectrum. Our results show that, while
any small amount of the anisotropy is enough to stabilize the domain walls against the
quantum fluctuations, no boundary condition exists that would stabilize a domain wall in
the isotropic model (Delta=1).