Finitely generated simple left orderable groups with vanishing second bounded cohomology

Abstract

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

Braided Thompson groups with and without quasimorphisms

“…The combination of these properties is interesting because it shows that in the celebrated Witte Morris Theorem [58] the hypothesis of amenability cannot be weakened to the vanishing of second bounded cohomology. The first finitely generated examples were found in [29]: this was the first step towards finding examples with the additional property of being nonindicable [28], answering a question of Navas [52]. Since bV is indicable, the existence of type F ∞ examples with these stronger properties is still open.…”

“…The most straightforward example is probably Thompson's group T , which has no quasimorphisms by virtue of being uniformly perfect (and even uniformly simple, see e.g., [41]). In fact, when the groups are even leftorderable, many of them have vanishing second bounded cohomology [29,28], and sometimes even vanishing bounded cohomology in every positive degree [48]. As a remark, since the examples coming from the procedure in [23] act on the plane by fixing a radial coordinate and acting by rotations, which is really a "one-dimensional" picture, one can view bV as providing the first truly "two-dimensional" example, i.e., one involving genuine braids.…”