The Addition Theorem for the Algebraic Entropy of Torsion Nilpotent Groups

TL;DR

This paper proves the Addition Theorem for algebraic entropy of endomorphisms in torsion nilpotent groups of any class, extending previous results and exploring applications to automorphisms of locally finite groups.

Contribution

It generalizes the Addition Theorem to all torsion nilpotent groups, including those of arbitrary nilpotency class, and provides new applications and reduction principles.

Findings

Addition Theorem holds for torsion nilpotent groups of any class.

Entropy of endomorphisms in torsion nilpotent groups is either infinite or a logarithm of a positive integer.

The theorem applies to automorphisms of locally finite and -hypercentral groups.

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 $\phi\in\operatorname{End}(G)$ either the entropy is infinite or $h(\phi)=\log(\alpha)$ for a positive integer $\alpha$. We further obtain, for automorphisms of locally finite groups, the Addition Theorem with respect to every term of the upper central series; in particular, it holds for automorphisms of $\omega$-hypercentral groups. Finally, we establish a reduction principle: if $\mathfrak X$ is a class of locally finite groups closed under taking subgroups and quotients, then the Addition Theorem for endomorphisms holds in $\mathfrak X$ if and only if it holds for locally finite groups generated by bounded sets.