Braided Thompson groups with and without quasimorphisms
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arXiv
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| Format: | Preprint |
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2022
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| author | Fournier-Facio, Francesco Lodha, Yash Zaremsky, Matthew C. B. |
| author_facet | Fournier-Facio, Francesco Lodha, Yash Zaremsky, Matthew C. B. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2204_05272 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Braided Thompson groups with and without quasimorphisms Fournier-Facio, Francesco Lodha, Yash Zaremsky, Matthew C. B. Group Theory Geometric Topology 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. |
| title | Braided Thompson groups with and without quasimorphisms |
| topic | Group Theory Geometric Topology |
| url | https://arxiv.org/abs/2204.05272 |