论文标题
直接产品，重叠的动作和关键的规律性
Direct products, overlapping actions, and critical regularity
论文作者
论文摘要
我们解决了计算该间隔的同构群的关键规律性的问题。 Our main result is that if $H$ and $K$ are two non-solvable groups then a faithful $C^{1,τ}$ action of $H\times K$ on a compact interval $I$ is {\em not overlapping} for all $τ>0$, which by definition means that there must be non-trivial $h\in H$ and $k\in K$ with disjoint support.作为推论，我们证明了右角artin组$（f_2 \ times f_2）*\ mathbb {z} $具有关键的规律性，也就是说，它承认忠实的$ c^1 $在$ i $上付出了$ i $，但没有忠实的忠诚$ c^{1，τ} $ action。这是一组指数增长的第一个明确示例，该指数增长没有非亚伯次指数的生长亚组，其关键规律性是有限的，实现的，并且确切地知道。我们得到的另一个推论是，汤普森的$ f $不承认$ i $上的忠实$ c^1 $重叠的动作，因此$ f*\ mathbb {z} $是当地表明的不承认不忠实的$ c^1 $的新示例。
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 faithful $C^{1,τ}$ action of $H\times K$ on a compact interval $I$ is {\em not overlapping} for all $τ>0$, 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 $C^{1,τ}$ action. This is the first explicit example of a group of exponential growth which is without nonabelian subexponential growth subgroups, whose critical regularity is finite, achieved, and known exactly. Another corollary we get is that Thompson's group $F$ does not admit a faithful $C^1$ 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 $I$.
