Finite-state self-similar actions of nilpotent groups

TL;DR

This paper characterizes when finitely generated torsion-free nilpotent groups admit finite-state self-similar actions, linking these properties to the Jordan normal form of an associated automorphism of the group's Lie algebra.

Contribution

It provides a complete characterization of finite-state self-similar actions of nilpotent groups using the Jordan normal form of a related automorphism.

Findings

All actions are finite-state if the automorphism's Jordan form satisfies certain conditions.

Existence of finite-state actions depends on the Jordan normal form of the automorphism.

Characterization is achieved via the Lie algebra automorphism of the Mal'cev completion.

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 $G$.