Spectral criteria for extending partial left orders

We answer affirmatively two extension questions posed by Bodart, Ciobanu, and Metcalfe in 2024: every partial left order extends to a total left order on the group of positive rational affine transformations, and on every group ℤ² ⋊T ℤ with T ∈ SL₂(ℤ) and tr(T) > 2. Both follow from a metabelian criterion: it suffices that the group be torsion-free, its derived subgroup have finite rational rank, and all conjugation eigenvalues on that subgroup be positive real. We derive more general extension results from structural theorems of Dong and Bodart–Dong. For groups with a finite-rank nilpotent normal subgroup and abelian quotient, extension of arbitrary partial left orders is equivalent to extension of total left orders from subgroups. This connects the cyclic case with Pierce’s classical theorem. A unified spectral criterion allows a locally nilpotent derived subgroup: under local finiteness and positive spectrum modulo its center, full extension is equivalent to the requirement that a central eigenvalue whose positive power is positive real is itself positive real. The proofs explain how local affine certificates yield orders on the prescribed ambient group.

GitHub repository: https://github.com/eulogica/spectral-left-order-extension