Bounded cohomology and scl of verbal wreath products

Given a countable group A with trivial bounded cohomology for all semi-separable coefficients and any group normal upper Gamma Γ Γ , we prove that the bounded cohomology of certain restricted verbal wreath products normal upper Gamma wreath product Superscript Super Subscript upper W Baseline upper A Γ ≀ W A Γ W A vanishes in positive degrees, for all semi-separable coefficients. Examples of such verbal wreath products include the standard restricted wreath product (extending a recent result by Monod for lamplighters groups) as well as verbal wreath products arising from n -solvable, n -nilpotent and k -Burnside ( k equals 2 comma 3 comma 4 comma 6 k = 2 , 3 , 4 , 6 $k = 2,3,4,6$ ) verbal products. Moreover, we prove that the stable commutator length of any verbal wreath product normal upper Gamma wreath product Superscript Super Subscript upper W Baseline upper A Γ ≀ W A Γ W A vanishes. As an application, we show that every group of type upper F Subscript p F p Fₚ isometrically embeds into a group of type upper F Subscript p F p Fₚ with vanishing bounded cohomology in degree n greater than or equals 1 n ≥ 1 n≥ 1</jats:tex-ma