直积的自同构与单射映射：关于近可解极小极大群的性质
Automorphisms and monomorphisms of direct products of virtually solvable minimax groups
AI总结：
本文研究了有限生成近可解极小极大群直积的自同构与单射映射性质，证明了在特定条件下，每个单射映射可唯一分解为θ·ζ形式，其中θ将每个因子映射到排列后的因子，并且ζ是中心且非对角的。
摘要：
本文研究了有限生成近可解极小极大群直积Γ=Γ₁×⋯×Γᵣ的自同构和单射映射。在ℚ-代数包不可分解的假设下，证明了每个Γ的单射映射可唯一分解为φ=θ·ζ，其中θ将每个因子映射到排列后的因子，其ℚ-同构包相同，ζ是中心且非对角的。反之，每个这样的对定义了Γ的单射映射，且φ是自同构当且仅当θ是。此不可分解假设是尖锐的：我们证明它不能被弱化为因子的直接不可分解性。证明分为三步：首先建立有限维李代数和代数李代数的相应中心混合性质，然后是连通线性代数群，最后通过ℚ-代数包将这些结果转移到极小极大群。这扩展了已知的nilpotent情况，从自同构到单射映射，以及从有限生成无 torsion nilpotent 群到更广泛的有限生成近可解极小极大群。作为应用，我们刻画了co-Hopfian直积，并推导了Reidemeister数和Reidemeister谱的公式。
英文摘要：
This paper studies automorphisms and monomorphisms of direct products $Γ=Γ_1\times\cdots\timesΓ_r$ of finitely generated virtually solvable minimax groups, a class containing all virtually polycyclic groups. Under an indecomposability assumption on the $\mathbb Q$-algebraic hulls, we prove that every monomorphism of $Γ$ factorizes uniquely as $φ=θ\cdotζ$, where $θ$ sends each factor into a permuted factor with $\mathbb Q$-isomorphic hull and $ζ$ is central and off-diagonal. Conversely, every such pair defines a monomorphism of $Γ$, and $φ$ is an automorphism if and only if $θ$ is. This indecomposability assumption is sharp: we show it cannot be weakened to direct indecomposability of the factors. The proof proceeds in three steps: first by establishing the corresponding central mixing property for finite-dimensional Lie algebras and algebraic Lie algebras, then for connected linear algebraic groups, and finally by transferring these results to minimax groups via $\mathbb Q$-algebraic hulls. This extends the previously known nilpotent case both from automorphisms to monomorphisms and from finitely generated torsion-free nilpotent groups to the broader class of finitely generated virtually solvable minimax groups. As applications, we characterize co-Hopfian direct products and derive formulas for Reidemeister numbers and Reidemeister spectra.
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