Displacement techniques in bounded cohomology

TL;DR

This paper explores various displacement techniques in bounded cohomology, introducing a new property called commuting cyclic conjugates that ensures vanishing of bounded cohomology across all degrees and coefficients.

Contribution

It introduces the property of commuting cyclic conjugates as a new, widely applicable displacement technique in bounded cohomology, expanding the toolkit for proving vanishing results.

Findings

The new property guarantees vanishing of bounded cohomology in all positive degrees.

Implications and relationships among different displacement techniques are analyzed.

Counterexamples are provided to illustrate limitations of certain methods.

Abstract

Several algebraic criteria, reflecting displacement properties of transformation groups, have been used in the past years to prove vanishing of bounded cohomology and stable commutator length. Recently, the authors introduced the property of commuting cyclic conjugates, a new displacement technique that is widely applicable and provides vanishing of the bounded cohomology in all positive degrees and all dual separable coefficients. In this note we consider the most recent along with the by now classical displacement techniques and we study implications among them as well as counterexamples.