Automorphisms and monomorphisms of direct products of virtually solvable minimax groups
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911627558780928 |
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| author | Deré, Jonas Vandermeersch, Ken |
| author_facet | Deré, Jonas Vandermeersch, Ken |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_24167 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Automorphisms and monomorphisms of direct products of virtually solvable minimax groups Deré, Jonas Vandermeersch, Ken Group Theory 20E36 (Primary), 20F16, 20G20, 17B40 (Secondary) 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. |
| title | Automorphisms and monomorphisms of direct products of virtually solvable minimax groups |
| topic | Group Theory 20E36 (Primary), 20F16, 20G20, 17B40 (Secondary) |
| url | https://arxiv.org/abs/2602.24167 |