Bounded cohomology and scl of verbal wreath products

Abstract

We study the bounded cohomology and the stable commutator length of verbal wreath products $\Gamma \wr^{_W}A$, where $A$ has trivial bounded cohomology for a sufficiently large class of coefficients.\\ We prove that the stable commutator length always vanishes, and that the bounded cohomology vanishes in positive degrees for some such verbal wreath products; including the standard restricted wreath products (extending a recent result by Monod for lamplighters groups), as well as verbal wreath products arising from n-solvable, $n$-nilpotent, and $k$-Burnside $(k = 2, 3, 4, 6)$ verbal products.\ As an application, we show that every group of type $F_p$ isometrically embeds into a group of type $F_p$ with vanishing bounded cohomology in positive degrees for a large class of coefficients.