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

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 the so-called homological von Neumann--Day Problem, as discussed by Calegari. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Next, we provide the first examples of finitely generated left orderable groups that are not locally indicable, and yet have vanishing second bounded cohomology. This proves that a homological analogue of the Witte-Morris Theorem does not hold. Finally, we combine some of the aforementioned results to provide the first examples of manifolds whose simplicial volumes are algebraic and irrational. This is further evidence towards a conjecture of Heuer and L\"{o}h.