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 Löh.

We provide new computations in bounded cohomology: A group is boundedly acyclic if its bounded cohomology with trivial real coefficients is zero in all positive degrees. We show that there exists a continuum of finitely generated non-amenable boundedly acyclic groups and that there exists a finitely presented boundedly acyclic group that is universal in the sense that it contains all finitely presented groups.On the other hand, we construct a continuum of finitely generated groups, whose bounded cohomology has uncountable dimension in all degrees greater than or equal to 2. Countable non-amenable groups with these two extreme properties were previously known to exist, but these constitute the first finitely generated examples.Finally, we show that various algorithmic problems on bounded cohomology are undecidable.

No abstract

A group is boundedly acyclic if its bounded cohomology with trivial real coefficients vanishes in all positive degrees. Amenable groups are boundedly acyclic, while the first nonamenable examples are the group of compactly supported homeomorphisms of $ {\mathbb {R}}^{n}$ (Matsumoto–Morita) and mitotic groups (Löh). We prove that binate (alias pseudo-mitotic) groups are boundedly acyclic, which provides a unifying approach to the aforementioned results. Moreover, we show that binate groups are universally boundedly acyclic. We obtain several new examples of boundedly acyclic groups as well as computations of the bounded cohomology of certain groups acting on the circle. In particular, we discuss how these results suggest that the bounded cohomology of the Thompson groups F, T, and V is as simple as possible.

We provide the first example of a finitely presented (in fact, type F ∞ ) group with elements whose stable commutator length is algebraic and irrational, answering a question of Calegari. Our example is the lift to the real line of the golden ratio Thompson group T τ : the circle analogue of the Cleary's golden ratio Thompson group F τ which acts on the interval. PreliminariesGroup actions will always be on the right. Accordingly, we will use the convention [g, h] = g −1 h −1 gh for commutators. Dynamics of group actions on 1-manifoldsGiven g ∈ Homeo + (M) for a given 1-manifold M, we define the support of g as:We identify S 1 with R/Z. Recall that Homeo + (R) and Homeo + (R/Z) are the groups of orientation preserving homeomorphisms of the real line and the circle. The latter group