The Addition Theorem for the Algebraic Entropy of Torsion Nilpotent 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.