Self-similarity of some soluble relatively free groups (Q2694002)

A group \(G\) is self-similar if the group has a faithful state-closed representation on an infinite regular one-rooted \(m\)-tree \(\mathcal{T}_{m}\) (\(m \geq 2\)); in addition, if \(G\) acts transitively on the first level of the tree, \(G\) is said to be transitive self-similar. The main results proved by the authors in the paper under review are the following. Theorem A: The free nilpotent group \(N_{r,c}\) of class \(c\) and finite rank \(r\) is recurrent transitive self-similar. Furthermore, \(N_{r,c}\) is an automata group. Theorem B: The free metabelian group \(M_{r}\) of rank \(r \geq 2\) is not transitive self-similar.