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 bV , which was introduced concurrently by Brin, has trivial second bounded cohomology.This makes bV the first example of a left-orderable group of type F ∞ that is not locally indicable and has trivial second bounded cohomology.This also makes bV an interesting example of a subgroup of the mapping class group of the plane minus a Cantor set that is non-amenable but has trivial second bounded cohomology, behaviour that cannot happen for finite-type mapping class groups.