  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).
