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
-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.