Braided Thompson groups with and without quasimorphisms

TL;DR

This paper investigates the quasimorphism properties and bounded cohomology of various braided Thompson groups, revealing diverse behaviors including infinite-dimensional spaces of quasimorphisms and trivial bounded cohomology in specific cases.

Contribution

It provides the first examples of braided Thompson groups with both infinite-dimensional and trivial second bounded cohomology, highlighting novel algebraic and geometric properties.

Findings

bV has infinite-dimensional space of quasimorphisms

rV and bF also have infinite-dimensional quasimorphism spaces

V has trivial second bounded cohomology, unlike other variants

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