Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

TL;DR

This paper develops criteria for vanishing second bounded cohomology in groups acting on 1-manifolds, leading to new results in group theory, geometry, and spectrum problems, including solutions to longstanding questions.

Contribution

It introduces a general criterion for vanishing second bounded cohomology, applies it to various groups, and constructs novel examples with specific geometric and algebraic properties.

Findings

Vanishing second bounded cohomology for certain nonamenable groups.

Existence of finitely generated simple left orderable groups with vanishing cohomology.

Examples of groups with irrational spectra of stable commutator length.

Abstract

We prove a general criterion for the vanishing of second bounded cohomology (with trivial real coefficients) for groups that admit an action satisfying certain mild hypotheses. This leads to new computations of the second bounded cohomology for a large class of groups of homeomorphisms of $1$-manifolds, and a plethora of applications. First, we demonstrate that the finitely presented and nonamenable group $G_0$ constructed by the second author with Justin Moore satisfies that every subgroup has vanishing second bounded cohomology. This provides the first solution to a homological version of the von Neumann--Day Problem, posed by Calegari. Next, we develop a technical refinement of our criterion to demonstrate the existence of finitely generated non-indicable (even simple) left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Navas. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Finally, we provide the first examples of manifolds whose simplicial volumes are algebraic and irrational, as further evidence towards a conjecture of Heuer and L\"{o}h.