We prove that the finitely generated simple left orderable groups constructed by the second author with Hyde have vanishing second bounded cohomology, both with trivial real and trivial integral coefficients. As a consequence, these are the first examples of finitely generated non-indicable left orderable groups with vanishing second bounded cohomology. This answers Question 8 from the 2018 ICM proceedings article of Andrés Navas.
Bounded cohomology and central extensionsWe will work with cohomology and bounded cohomology with trivial real coefficients, and use the definition in terms of the bar resolution. We refer the reader to [3] and [10,22] for a general and complete treatment of ordinary and bounded cohomology, respectively.For every n ≥ 0, denote by C n (G) the set of real-valued functions on G n . By convention, G 0 is a single point, so C 0 (G) ∼ = R consists only of constant functions. We define differential operators