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Automorphisms of direct products of virtually solvable minimax groups

Jonas Deré, Ken Vandermeersch

Abstract

This paper studies injective endomorphisms of direct products $Γ=Γ_1\times\cdots\times Γ_r$ of finitely generated virtually solvable minimax groups, a class that includes virtually polycyclic groups. If each factor has a Zariski connected, $\mathbb Q$-indecomposable algebraic hull, then any injective endomorphism of $Γ$ factors uniquely as $\varphi=θ\cdotζ$ where $θ$ permutes the factors up to $\mathbb Q$-isomorphism of hulls and $ζ$ is off-diagonal and central. Conversely, any such pair defines an injective endomorphism of the direct product. The proof passes to the $\mathbb Q$-algebraic hull of these groups, using a central mixing theorem for linear algebraic groups and their Lie algebras. As an application of the factorization, we characterize co-Hopfian direct products of such groups and compute Reidemeister numbers and spectra.

Automorphisms of direct products of virtually solvable minimax groups

Abstract

This paper studies injective endomorphisms of direct products of finitely generated virtually solvable minimax groups, a class that includes virtually polycyclic groups. If each factor has a Zariski connected, -indecomposable algebraic hull, then any injective endomorphism of factors uniquely as where permutes the factors up to -isomorphism of hulls and is off-diagonal and central. Conversely, any such pair defines an injective endomorphism of the direct product. The proof passes to the -algebraic hull of these groups, using a central mixing theorem for linear algebraic groups and their Lie algebras. As an application of the factorization, we characterize co-Hopfian direct products of such groups and compute Reidemeister numbers and spectra.
Paper Structure (12 sections, 11 theorems, 6 equations)

This paper contains 12 sections, 11 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Lie algebras and linear algebraic groups.
  3. Minimax groups.
  4. Applications.
  5. Notation.
  6. Preliminaries
  7. Linear algebraic K-groups in characteristic 0
  8. The Lie algebra of a K-group.
  9. The unipotent radical and tori.
  10. The determinant of a Lie algebra endomorphism
  11. K-algebraic Lie algebras
  12. A K-algebraic Krull-Schmidt theorem

Key Result

Theorem 1.2

Let $\mathcal{L} = \mathcal{L}_1 \oplus \mathcal{L}_2 \oplus \ldots \oplus \mathcal{L}_r$ be a direct sum of finite-dimensional, non-abelian, directly indecomposable Lie algebras over an arbitrary field. Then $\mathcal{L}$ satisfies $\mathsf{CMP}$ on automorphisms.

Theorems & Definitions (15)

  • Definition 1.1: Central Mixing Property
  • Theorem 1.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E
  • Corollary F
  • Definition 2.1: $K$-algebraic Lie subalgebras and morphisms
  • Theorem 2.2
  • ...and 5 more