Intransitive Self-similar Groups

Abstract

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 could not produce a faithful transitive self-similar representations for any $m\ge 2$ (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 ${\mathbb{Z}}^{\omega }$, $\mathbb{Z}\wr \mathbb{Z}$ and $(\mathbb{Z}\wr \mathbb{Z})\wr C_2$.