The Addition Theorem for the Algebraic Entropy of Torsion Nilpotent Groups
Menachem Shlossberg
TL;DR
This work proves the Addition Theorem for the algebraic entropy of endomorphisms on torsion nilpotent groups of arbitrary nilpotency class, extending the known class-2 result. It derives a dichotomy for entropy on such groups: a finite entropy must be of the form $h(\phi)=\log(\alpha)$ with $\alpha\in\mathbb{N}$, and establishes AT for automorphisms of locally finite groups with respect to all terms of the upper central series. Leveraging a reduction principle, the authors demonstrate that AT in a variety of locally finite groups reduces to AT for locally finite, bounded-generated subgroups. They further extend the AT framework to locally finite and $\omega$-hypercentral groups, providing a unified approach to entropy behavior through central extensions and upper central series, with broad structural implications for non-abelian group dynamics.
Abstract
The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved by Dikranjan, Goldsmith, Salce and Zanardo. It was later extended by Shlossberg to torsion nilpotent groups of class 2. As our main result, we prove the Addition Theorem for endomorphisms of torsion nilpotent groups of arbitrary nilpotency class. As an application, we show that if $G$ is a torsion nilpotent group, then for every $φ\in \mathrm{End}(G)$ either the entropy $h(φ)$ is infinite or $h(φ)=\log(α)$ for some $α\in\mathbb N$. We further obtain, for automorphisms of locally finite groups, the Addition Theorem with respect to all terms of the upper central series; in particular, the Addition Theorem holds for automorphisms of $ω$-hypercentral groups. Finally, we establish a reduction principle: if $\mathfrak X$ is a variety of locally finite groups, then the Addition Theorem for endomorphisms holds in $\mathfrak X$ if and only if it holds for locally finite groups generated by bounded sets.