  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.

Group Theory;Dynamical Systems;Geometric Topology