论文标题
不及物的自相似群体
Intransitive Self-similar Groups
论文作者
论文摘要
据说一个小组是自相似的，只要它承认某些常规$ m $树的忠实国家封闭式表示形式，并且据说该小组是瞬息的自相似性，此外还可以在树的第一层引起及物动词。构建组的及物相似表示的标准方法是通过所讨论组的单个虚拟内态性的。最近，结果表明，将这种方法应用于受限的花环产品$％\ mathbb {z} \ wr \ mathbb {z} $无法为任何$ m \ geq 2 $（参见，\ cite {ds}}）产生忠实的透性自相似表示。在这项工作中，我们研究了封闭状态的表示，而不假设具有传递性条件。该一般动作被转化为一组虚拟的内态性，与树的第一层上的动作的不同轨道相对应。通过这种方式，我们为许多组提供了忠实的自相似表示，其中一些也是有限状态，例如$ \ Mathbb {z}^ω$，$ \ Mathbb {z} \ wr \ wr \ wr \ mathbb {z} $ and $（\ wr \ wr \ wr \ wr \ mathbb} $}
A group is said to be self-similar provided it admits a faithful state-closed representation on some regular $m$-tree and the group is said to be transitive self-similar provided additionally it induces transitive action on the first level of the tree. A standard approach for constructing a transitive self-similar representation of a group has been by way of a single virtual endomorphism of \ the group in question. Recently, it was shown that this approach when applied to the restricted wreath product $% \mathbb{Z}\wr \mathbb{Z}$ could not produce a faithful transitive self-similar representations for any $m\geq 2$ (see, \cite{DS}). In this work we study state-closed representations without assuming the transitivity condition. This general action is translated into a set of virtual endomorphisms corresponding to the different orbits of the action on the first level of the tree. In this manner, we produce faithful self-similar representations, some of which are also finite-state, for a number of groups such as $\mathbb{Z}^ω$, $\mathbb{Z}\wr \mathbb{Z}$ and $(\mathbb{Z} \wr \mathbb{Z}) \wr C_{2}$.
