Homological Finiteness Properties of Wreath Products

honimichl j.jak;

We study the homological finiteness properties FPm of wreath products  = H X G. We show that, when H has infinite abelianization,  is of type FPm if and only if both G and H have type
FPm and G acts (diagonally) on Xi with stabilizers of type FPm−i and with finitely many orbits for all 1 ≤ i ≤ m. If furthermore H is torsion-free, we give a criterion for  to be Bredon-FPm with respect to the class of finite subgroups of . Finally, when H has infinite abelianization and χ :  → R is a non-zero homomorphism with
χ(H) = 0, we classify when [χ] belongs to the Bieri–Neumann–Strebel–Renz invariantm(,Z).