论文标题
有或没有编织的编织汤普森群体
Braided Thompson groups with and without quasimorphisms
论文作者
论文摘要
我们研究了汤普森组的各种编织版本的准晶体和有界的共同体学。我们的第一个主要结果是，Brin-Dehornoy编织的Thompson Group $ bv $具有无限尺寸的准牙作用空间，因此无限二维第二有界的共同体学。这意味着，尽管$ BV $是完美的，但与Thompson的Group $ V $相反，$ BV $并不统一。我们还证明，$ bv $的亲戚，例如丝带编织的汤普森集团$ rv $和纯编织的汤普森集团$ bf $，类似地具有无限尺寸的准牙作用空间。我们的第二个主要结果是，与之形成鲜明对比的是，$ bv $的近亲表示$ \ hat {bv} $，该$由Brin同时引入，具有微不足道的第二界共同体。这使$ \ hat {bv} $成为左顺序类型$ \ operatatorName {f} _ \ infty $的第一个示例，该组不是本地指示的，并且具有微不足道的第二个有限的共同体。这也使$ \ hat {bv} $成为平面映射类组的子组的有趣示例。
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.
