The Addition Theorem for two-step nilpotent torsion groups

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved in [2]. Later, this result was extended to all abelian groups [1] and, recently, to all torsion finitely quasihamiltonian groups [3]. In contrast, when it comes to metabelian groups, the additivity of the algebraic entropy fails [4]. Continuing the research within the class of locally finite groups, we prove that the Addition Theorem holds for two-step nilpotent torsion groups.

[1] D. Dikranjan, A. Giordano Bruno, Entropy on abelian groups, Adv. Math. 298 (2016) 612–653.
[2] D. Dikranjan, B. Goldsmith, L. Salce, P. Zanardo, Algebraic entropy of endomorphisms of abelian groups, Trans. Amer. Math. Soc. 361 (2009) 3401–3434.
[3] A. Giordano Bruno, F. Salizzoni, Additivity of the algebraic entropy for locally finite groups with permutable finite subgroups, J. Group Theory 23 (2020), 831–846 DOI 10.1515 / jgth-2019-0096.
[4] A. Giordano Bruno, P. Spiga, Some properties of the growth and of the algebraic entropy of group endomorphisms, J. Group Theory 20 (2017), no. 4, 763–774.