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Bounded cohomology and scl of verbal wreath products

Elena Bogliolo

TL;DR

This work extends Monod's bounded cohomology framework to verbal wreath products by introducing semi-separable coefficient modules 𝔛^{ssep} and leveraging Hochschild-Serre spectral sequences to prove bounded acyclicity without requiring amenability of the acting group A. The main result shows that Γ ⧸_X A is 𝔛^{ssep}-boundedly n-acyclic whenever A is 𝔛^{ssep}-boundedly n-acyclic and A has infinite orbits on X, enabling a broad class of new examples where H_b^k(Γ;V)=0 for 1 ≤ k ≤ n for semi-separable V. This framework is then extended to verbal wreath products, yielding vanishing bounded cohomology for restricted j-nilpotent, j-solvable, and k-Burnside wreaths and providing amenable projections that transfer acyclicity. Applications include proving the vanishing of stable commutator length for these wreath products and constructing isometric embeddings of arbitrary groups into 𝔛^{ssep}-boundedly acyclic groups while preserving finiteness properties, with Thompson F playing a key role in the embedding constructions. Overall, the paper broadens the landscape of groups with vanishing bounded cohomology and connects bounded cohomology to structural group operations such as verbal wreath products and stability phenomena in scl.

Abstract

We study the bounded cohomology and the stable commutator length of verbal wreath products $Γ\wr^{_W}A$, 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.

Bounded cohomology and scl of verbal wreath products

TL;DR

This work extends Monod's bounded cohomology framework to verbal wreath products by introducing semi-separable coefficient modules 𝔛^{ssep} and leveraging Hochschild-Serre spectral sequences to prove bounded acyclicity without requiring amenability of the acting group A. The main result shows that Γ ⧸_X A is 𝔛^{ssep}-boundedly n-acyclic whenever A is 𝔛^{ssep}-boundedly n-acyclic and A has infinite orbits on X, enabling a broad class of new examples where H_b^k(Γ;V)=0 for 1 ≤ k ≤ n for semi-separable V. This framework is then extended to verbal wreath products, yielding vanishing bounded cohomology for restricted j-nilpotent, j-solvable, and k-Burnside wreaths and providing amenable projections that transfer acyclicity. Applications include proving the vanishing of stable commutator length for these wreath products and constructing isometric embeddings of arbitrary groups into 𝔛^{ssep}-boundedly acyclic groups while preserving finiteness properties, with Thompson F playing a key role in the embedding constructions. Overall, the paper broadens the landscape of groups with vanishing bounded cohomology and connects bounded cohomology to structural group operations such as verbal wreath products and stability phenomena in scl.

Abstract

We study the bounded cohomology and the stable commutator length of verbal wreath products , where 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, -nilpotent, and -Burnside verbal products.\ As an application, we show that every group of type isometrically embeds into a group of type with vanishing bounded cohomology in positive degrees for a large class of coefficients.
Paper Structure (12 sections, 23 theorems, 72 equations)

This paper contains 12 sections, 23 theorems, 72 equations.

Table of Contents

  1. Preliminaries
  2. Coefficient modules
  3. Bounded cohomology
  4. Semi-separable modules
  5. Group actions
  6. Zimmer-amenable actions
  7. Ergodic actions
  8. Bounded acyclicity of permutational wreath product
  9. Examples
  10. Verbal products
  11. Application to stable commutator length
  12. Application to embeddings

Key Result

Theorem 1

Let $\Gamma$ and $A$ be discrete groups. Let $n\geq 1$ and suppose that $A$ is $\mathfrak{X}^{\textup{ssep}}$-boundedly $n$-acyclic and countable. If $X$ is a countable $A$-set such that all the $A$-orbits are infinite, then, the restricted permutational wreath product $\Gamma\wr_X A$ is $\mathfrak{

Theorems & Definitions (78)

  • Theorem 1: Theorem \ref{['acyclicwr']}
  • Corollary 2: Corollary \ref{['verbalcase']}
  • Theorem 3: Theorem \ref{['sclcase']}
  • Theorem 4: Theorem \ref{['finitenesscase']}
  • Definition 1.1: normed $\Gamma$-module
  • Definition 1.2: coefficient $\Gamma$-module Mon
  • Definition 1.3: separable module
  • Remark 1.4
  • Remark 1.5
  • Definition 1.6
  • ...and 68 more