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We prove a general criterion for the vanishing of second bounded cohomology(with trivial real coefficients) for groups that admit an action satisfyingcertain mild hypotheses. This leads to new computations of the second boundedcohomology for a large class of groups of homeomorphisms of $1$-manifolds, anda plethora of applications. First, we demonstrate that the finitely presentedand nonamenable group $G_0$ constructed by the second author with Justin Mooresatisfies that every subgroup has vanishing second bounded cohomology. Thisprovides the first solution to a homological version of the von Neumann--DayProblem, posed by Calegari. Next, we develop a technical refinement of ourcriterion 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 weprovide the first examples of finitely presented groups whose spectrum ofstable commutator length contains algebraic irrationals, answering a questionof Calegari. Finally, we provide the first examples of manifolds whosesimplicial volumes are algebraic and irrational, as further evidence towards aconjecture of Heuer and L\"{o}h.

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