Braided Thompson groups with and without quasimorphisms

We study quasimorphisms and bounded cohomology of a variety of braided versions of Thompson groups. Our first main result is that the Brin-Dehornoy braided Thompson group bV has an infinite -dimensional space of quasimorphisms and thus infinite -dimensional second bounded cohomology. This implies that, despite being perfect, bV is not uniformly perfect, in contrast to Thompson's group V . We also prove that relatives of bV like the ribbon braided Thompson group rV and the pure braided Thompson group bF similarly have an infinite -dimensional space of quasimorphisms. Our second main result is that, in stark contrast, the close relative of bV denoted by c bV , which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes c bV the first example of a left -orderable group of type F 1 that is not locally indicable and has trivial second bounded cohomology. This also makes c bV an interesting example of a subgroup of the mapping class group of the plane minus a Cantor set that is nonamenable but has trivial second bounded cohomology, behavior that cannot happen for finite -type mapping class groups.