Intransitive Self-similar Groups

2020/04/19 by Dantas, Alex C., Santos, Tulio M. G., Sidki, Said N. · 2 citations
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2004.08941

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 % ℤ\wr ℤ could not produce a faithful transitive self-similar representations for any m≥ 2 (see, \citeDS). 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 ℤ^ω, ℤ\wr ℤ and (ℤ \wr ℤ) \wr C₂.

Cited by

• On self-similarity of finitely generated torsion-free nilpotent groups
• Non self-similar metabelian groups

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