Finite-state self-similar actions of nilpotent groups

Abstract

Let G be a finitely generated torsion-free nilpotent group and \({\phi:H\rightarrow G}\) be a surjective homomorphism from a subgroup H < G of finite index with trivial \({\phi}\) -core. For every choice of coset representatives of H in G there is a faithful self-similar action of the group G associated with \({(G, \phi)}\). We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for \({(G, \phi)}\). These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism \(\widehat{\phi}\) of the Lie algebra of the Mal’cev completion of G.