Direct products, overlapping actions, and critical regularity
Direct products, overlapping actions, and critical regularity
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DOI：
10.3934/jmd.2021009
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发表时间：
2020-10
影响因子：
1.1
通讯作者：
Sang-hyun Kim;T. Koberda;C. Rivas
中科院分区：
文献类型：
--
作者：
Sang-hyun Kim;T. Koberda;C. Rivas
讨论了区间上同胚群的临界正则性的计算问题。我们的主要结果是，如果$H$和$K$是两个不可解的群，那么$H$乘以$K$在紧区间$I$上的$C^1$作用不可能是{\em重叠}的，这就意味着在不相交的支持下，$H$和$K$中必定存在非平凡的$H$和$K$。作为一个推论，我们证明了直角Artin群$(F_2\乘以F_2)*\mathbb{Z}$具有临界正则性1，即它对$I$有忠实的$C^{1,\tau}$作用，但对$\tau> $没有忠实的$C^{1,\tau}$作用。这是一组指数增长的第一个明确的例子，其临界规律性是有限的，确切地知道并得到的。另一个推论是Thompson的群$F$不允许$C^1$重叠作用于$I$，因此$F*\mathbb{Z}$是局部可指示群不允许$C^1$忠实作用于$I$的新例子。
We address the problem of computing the critical regularity of groups of homeomorphisms of the interval. Our main result is that if $H$ and $K$ are two non-solvable groups then a $C^1$ actions of $H\times K$ on a compact interval $I$ cannot be {\em overlapping}, which by definition means that there must be non-trivial $h\in H$ and $k\in K$ with disjoint support. As a corollary we prove that the right-angled Artin group $(F_2\times F_2)*\mathbb{Z}$ has critical regularity one, which is to say that it admits a faithful $C^1$ action on $I$, but no faithful $C^{1,\tau}$ action for $\tau>0$. This is the first explicit example of a group of exponential growth whose critical regularity is finite, known exactly, and achieved. Another corollary we get is that Thompson's group $F$ does not admit a $C^1$ overlapping action on $I$, so that $F*\mathbb{Z}$ is a new example of a locally indicable group admitting no faithful $C^1$--action on $I$.