自由积与同构群的代数结构
Free products and the algebraic structure of diffeomorphism groups
摘要：
设$M$是一个紧致的单流形，$\mathrm{Diff}^{1+\mathrm{bv}}(M)$表示$M$的一阶导数有界变差的保$C^1$定向同构群.我们证明，如果$G$是一个群体，这不是虚拟的亚阿贝尔，那么$(G\times\mathbb{Z})*\mathbb{Z}$是不实现为一个子群的$\mathrm{Diff}^{1+\mathrm{bv}}(M)$。这给出了第一个示例，其中生成的组$G,H\le \mathrm{Diff}_+^\infty(M)$使得$G\ast H$不嵌入$\mathrm{Diff}^{1+\mathrm{bv}}(M)$。相反，对于所有可数群$G,H\le\mathrm{Homeo}^+(M)$，存在嵌入$G\ast H\to \mathrm{Homeo}^+(M)$。我们推导出许多同胚的公共群不嵌入到$\mathrm{Diff}^{1+\mathrm{bv}}(M)$中，例如$\mathbb{Z}$与Thompson群$F$的自由积。我们还完成了直角Artin群的分类，它可以顺利地作用于$M$，特别是，恢复的主要结果的作者与Baik的联合工作。也就是说，一个直角Artin群$A(Γ)$要么允许$M$上有忠实的$C^{\infty}$作用，要么$A(Γ)$不允许$M$上有忠实的$C^{1+\mathrm{bv}}$作用。在前一种情况下，是$A(Γ)\cong\prod_i G_i$，其中$G_i$是自由阿贝尔群的自由积。最后，我们开发了一个层次的直角Artin群，与层次对应的数量半共轭类的可能行动，这些群体的$S^1$。
英文摘要：
Let $M$ be a compact one--manifold, and let $\mathrm{Diff}^{1+\mathrm{bv}}(M)$ denote the group of $C^1$ orientation preserving diffeomorphisms of $M$ whose first derivatives have bounded variation. We prove that if $G$ is a group which is not virtually metabelian, then $(G\times\mathbb{Z})*\mathbb{Z}$ is not realized as a subgroup of $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. This gives the first examples of finitely generated groups $G,H\le \mathrm{Diff}_+^\infty(M)$ such that $G\ast H$ does not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$. By contrast, for all countable groups $G,H\le\mathrm{Homeo}^+(M)$ there exists an embedding $G\ast H\to \mathrm{Homeo}^+(M)$. We deduce that many common groups of homeomorphisms do not embed into $\mathrm{Diff}^{1+\mathrm{bv}}(M)$, for example the free product of $\mathbb{Z}$ with Thompson's group $F$. We also complete the classification of right-angled Artin groups which can act smoothly on $M$ and in particular, recover the main result of a joint work of the authors with Baik. Namely, a right-angled Artin group $A(Γ)$ either admits a faithful $C^{\infty}$ action on $M$, or $A(Γ)$ admits no faithful $C^{1+\mathrm{bv}}$ action on $M$. In the former case, $A(Γ)\cong\prod_i G_i$ where $G_i$ is a free product of free abelian groups. Finally, we develop a hierarchy of right-angled Artin groups, with the levels of the hierarchy corresponding to the number of semi-conjugacy classes of possible actions of these groups on $S^1$.
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