The Addition Theorem for the Algebraic Entropy of Nilpotent Groups

Abstract

The Addition Theorem for the algebraic entropy of group endomorphisms was first proved for torsion abelian groups by Dikranjan, Goldsmith, Salce and Zanardo, and was later extended to all abelian groups by Dikranjan and Giordano Bruno. Shlossberg subsequently proved it for torsion nilpotent groups of class 2. As our main result, we prove the Addition Theorem for endomorphisms of nilpotent groups of arbitrary nilpotency class, without any torsion assumption. In the torsion case, this yields in particular that, if G is a torsion nilpotent group and \phi\in\operatorname{End}(G), then either h(\phi)=\infty or h(\phi)=\log(\alpha) for some positive integer \alpha. We further prove that, for every group G and every automorphism \phi\in\operatorname{Aut}(G), the Addition Theorem holds with respect to each term of the upper central series. As a consequence, the Addition Theorem holds for automorphisms of every \omega-hypercentral group. 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 holds for endomorphisms of groups in \mathfrak X if and only if it holds for endomorphisms of those groups in \mathfrak X generated by bounded subsets, that is, subsets whose element orders are uniformly bounded.