Finite-state self-similar actions of nilpotent groups

Let $G$ be a finitely generated torsion-free nilpotent group and $ϕ:H\rightarrow G$ be a surjective homomorphism from a subgroup $H<G$ of finite index with trivial $ϕ$-core. For every choice of coset representatives of $H$ in $G$ there is a faithful self-similar action of the group $G$ associated with $(G,ϕ)$. 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 $(G,ϕ)$. These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism $\widehatϕ$ of the Lie algebra of the Mal'cev completion of $G$.

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