Braided Thompson groups with and without quasimorphisms

Abstract

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 $\widehat{bV}$, which was introduced concurrently by Brin, has trivial second bounded cohomology. This makes $\widehat{bV}$ the first example of a left-orderable group of type $F_{\infty }$ that is not locally indicable and has trivial second bounded cohomology. This also makes $\widehat{bV}$ 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.

v2: final version, to appear in Algebraic & Geometric Topology

References in corpus (4)

• Amenable groups that act on the line
• Bounded cohomology and binate groups
• Asymptotically rigid mapping class groups I: Finiteness properties of braided Thompson's and Houghton's groups
• Braiding groups of automorphisms and almost-automorphisms of trees

Cited by in corpus (1)

• Infinite-type loxodromic isometries of the relative arc graph