论文标题
康托尔的一组同态性的足够条件
Sufficient conditions for a group of homeomorphisms of the Cantor set to be two-generated
论文作者
论文摘要
令$ \ mathfrak {c} $为一些cantor空间。我们研究了有力的$ \ mathfrak {c} $的同态形态，或者是完美无瑕的，我们在此处介绍这两个术语。 我们说一个组$ g \ leq \ leq \ peratatorName {homo}（\ mathfrak {c}）$是$ growter $，如果对于任何clopen set set set $ a $ a $ a $ a $ and $ b $ b $ b $ a $ b $ a $ a $ a $ a $ a $ a $ c $γ\ in $ f $ b的$ \ mathfrak a $ \ bets c} c} C $。剧烈的影响与爱泼斯坦提出的某些条件相似，证明某些空间同态同态具有简单的换向器亚组（和/或相关条件，如Matui或Ling的某些工作中所提出的）。 一个非平凡的$ g \ leq \ leq \ operatorName {homeo}（\ mathfrak {c}）$是$ $ $ $ $，如果对于所有$ k $和$ k $和$ w $ a a y non-k $变量上的无人物自由降低的产品表达（包括在$ k $ sumplass）上（包括副符号），一个特定的子级$ w（g），$ w（g）c $ comp $ comp $ compl $ compl $ compl $ compl $ compl $ compr of corval $ compl $ w。例如，确实，完美的群体既完美又是违法的。 We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) vigorous groups are simple if and only if they are flawless, and, 3) the class of vigorous simple subgroups of $\operatorname{Homeo}(\mathfrak{C})$ is fairly broad (it contains many well known groups such as the commutator subgroups of the Higman-Thompson groups $ g_ {n，r} $，Brin-Thompson组$ NV $，Röver's$ V（γ）$以及Nekrashevych的其他“简单的动态起源组”组，并且该类在各种自然结构下都关闭）。
Let $\mathfrak{C}$ be some Cantor space. We study groups of homeomorphisms of $\mathfrak{C}$ which are vigorous, or, which are flawless, where we introduce both of these terms here. We say a group $G\leq \operatorname{Homeo}(\mathfrak{C})$ is $vigorous$ if for any clopen set $A$ and proper clopen subsets $B$ and $C$ of $A$ there is $γ\in G$ in the pointwise-stabiliser of $\mathfrak{C}\backslash A$ with $Bγ\subseteq C$. Being vigorous is similar in impact to some of the conditions proposed by Epstein in his proof that certain groups of homeomorphisms of spaces have simple commutator subgroups (and/or related conditions, as proposed in some of the work of Matui or of Ling). A non-trivial group $G\leq \operatorname{Homeo}(\mathfrak{C})$ is $flawless$ if for all $k$ and $w$ a non-trivial freely reduced product expression on $k$ variables (including inverse symbols), a particular subgroup $w(G)_\circ$ of the verbal subgroup $w(G)$ is the whole group. It is true, for instance, that flawless groups are both perfect and lawless. We show: 1) simple vigorous groups are either two-generated by torsion elements, or not finitely generated, 2) vigorous groups are simple if and only if they are flawless, and, 3) the class of vigorous simple subgroups of $\operatorname{Homeo}(\mathfrak{C})$ is fairly broad (it contains many well known groups such as the commutator subgroups of the Higman-Thompson groups $G_{n,r}$, the Brin-Thompson groups $nV$, Röver's group $V(Γ)$, and others of Nekrashevych's `simple groups of dynamical origin', and, the class is closed under various natural constructions).
