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arXiv 2601.09643math.GR
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torsion nilpotent 群的代数熵加法定理

The Addition Theorem for the Algebraic Entropy of Nilpotent Groups

Reichman University（莱希曼大学）

Menachem Shlossberg

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9 pages
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AI总结：

本文证明了torsion nilpotent群的代数熵加法定理，并探讨了其在locally finite群中的应用及减缩原理。

摘要：

torsion nilpotent 群的代数熵加法定理被Dikranjan、Goldsmith、Salce和Zanardo证明。后来，Shlossberg将其扩展到nilpotency class为2的torsion nilpotent群。我们的主要结果是证明了torsion nilpotent群的任意nilpotency class的加法定理。作为应用，我们证明如果G是torsion nilpotent群，则对于每一个ϕ∈End(G)，熵要么是无限的，要么h(ϕ)=log(α)，其中α是正整数。我们进一步获得，对于locally finite群的自同构，关于上中央列的每一个项的加法定理；特别是，它适用于ω-hypercentral群的自同构。最后，我们建立了一个减缩原理：如果X是locally finite群的类，且在取子群和商群下闭合，则endomorphisms的加法定理在X中成立当且仅当在locally finite群由有界集合生成时成立。

英文摘要：

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 ϕ\in\operatorname{End}(G), then either h(ϕ)=\infty or h(ϕ)=\log(α) for some positive integer α. We further prove that, for every group G and every automorphism ϕ\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 ω-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.

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