  We present a full description of the Bieri-Neumann-Strebel invariant of
restricted permutational wreath products of groups. We also give partial
results about the 2-dimensional homotopical invariant of such groups. These
results may be turned into a full picture of these invariants when the
abelianization of the basis group is infinite. We apply these descriptions to
the study of the Reidemeister number of automorphisms of wreath products in
some specific cases.
