Bounded cohomology and scl of verbal wreath products

TL;DR

This paper investigates the bounded cohomology and stable commutator length of verbal wreath products, showing vanishing results and embedding properties that extend previous work on lamplighter groups and specific algebraic structures.

Contribution

It proves the vanishing of stable commutator length and bounded cohomology in positive degrees for a broad class of verbal wreath products, extending known results.

Findings

Stable commutator length always vanishes.

Bounded cohomology vanishes in positive degrees for certain verbal wreath products.

Groups of type F_p can embed into groups with vanishing bounded cohomology.

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.