Subject / 16 records

Ordered groups

Left orders, bi-orders, positive identities, and lattice-ordered groups.

Classification: Second Brain

16 records

• 10.5281/zenodo.238380122026

  Order extension in Chehata's group

  Martino Lupini · Orderability, Groups

  In the real case, Chehata's group consists of the increasing piecewise-linear homeomorphisms of the real line with finitely many pieces and bounded support. We prove that every bi-invariant partial order on this group extends to a bi-invariant total order, answering the real-field form of the question recorded in the 2009 survey of Bludov, Glass, Kopytov, and Medvedev. More precisely, a set of nonidentity elements generates a normal semigroup containing the identity if and only if the logarithms of its initial slopes and those of its final slopes each admit a nonempty positive integer relation. Every partial order therefore extends to an endpoint-slope order induced by an additive total order on the abstract group of real numbers. These orders form two disjoint families and exhaust all bi-orders; their number is 2^{2^{aleph_0}}. The proof creates an arbitrary endpoint germ after cancellation of a positive germ relation and then applies an equal-germ positive factorization argument. Over an Archimedean ordered field F, the analogous criterion and classification use additive orders on the logarithmic slope group and give 2^{|F|} bi-orders.

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• 10.5281/zenodo.238353112026

  Groups with prescribed finite numbers of lattice orders

  Martino Lupini · Orderability, Groups

  We prove that the possible finite numbers of lattice group orders on an ℓ-orderable group are exactly 1 and the positive even integers. We also construct, for every n ≥ 1, a countable nonabelian simple group with exactly 2n bi-invariant total orders. These results answer the finite-spectrum questions in the 2009 problem list of Bludov, Glass, Kopytov, and Medvedev. The lattice-order construction rests on a classification of all lattice orders on certain groups of homeomorphisms of the line. For the complementary examples, we show that every lattice order on the affine family used here, due to Kopytov, is total.

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• 10.5281/zenodo.238280862026

  Normal-semigroup generators for the bi-order cones of Thompson's group F′

  Martino Lupini · Thompson groups, Orderability, Groups

  The commutator subgroup F′ of Thompson's group has four bi-invariant total orders, determined by the slope at the first or last point of support. We prove that the positive cone of each order is generated by two elements as a normal semigroup: every positive element is a nonempty product of conjugates of those two elements. An explicit pair f, g works for all four cones, using the four choices of signs (f, g), (f⁻¹, g⁻¹), (f⁻¹, g), and (f, g⁻¹). This answers the finite-generation question posed by Navas and Rivas in 2008, including their stronger requirement that the same pair yield both prescribed partitions of F′\{1}. Two generators are necessary for each cone.

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• 10.5281/zenodo.238040062026

  Order-universal free subgroups and bi-order extension

  Martino Lupini · Orderability, Groups

  We construct fixed free subgroups along which every prescribed bi-order extends to the ambient group. Such subgroups exist inside every nontrivial normal subgroup of a free group of rank two. A finite-support localization theorem transfers this property to other bi-orderable groups, using one visible element and an invariant order on the actual intermediate quotient. We apply it to every bi-orderable rank-two free-by-cyclic group, to higher-rank monodromies with positive real homology spectrum not entirely equal to one, and to an explicit IA mapping torus. The last example and all its height cyclic covers admit no compatible cyclic-cover free quotient in any integral character direction, but still have the required universal free subgroups. Consequently total Bi-order Extension is undecidable in each of these fixed groups, already for a fixed finite background and one varying word. The argument also distinguishes the order-transfer problem from the existence of free quotients in finite covers.

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• 10.5281/zenodo.236405402026

  Positive normal identities and bi-order extension

  Martino Lupini · Orderability, Groups

  We prove that it is undecidable whether the identity in a free group of any fixed finite rank at least two is a nonempty product of conjugates of given elements, with only positive occurrences of those elements allowed. It is also undecidable whether a finite subset of that group can be made positive in a total order invariant under multiplication on both sides. Both conclusions hold for a fixed finite background together with one varying word, all lying in the commutator subgroup. These give negative answers to the normal identity and bi-order questions stated online in 2025 by Bodart, Ciobanu and Metcalfe, and to the bi-order extension question formulated by Colacito and Metcalfe (2017). The second result also settles their 2019 question about the word problem for free representable lattice-ordered groups: it is undecidable already on two generators.

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• 10.5281/zenodo.235230912026

  Simple left-orderable groups with prescribed finiteness length

  Martino Lupini · Orderability, Groups

  For every integer n ≥ 2 we construct an infinite simple left-orderable group of type F_n and not of type FP_{n+1}(ℤ), and hence not of type F_{n+1}, where type F_n means that a classifying space has finite n-skeleton. This answers affirmatively the left-orderable interpretation of a question of Xiaolei Wu in the problem list associated with the 2024 CIRM meeting. The construction inserts arithmetic groups of one-sided germs into a group of homeomorphisms of the line. A forest complex controls the finiteness of the germ groups, and an actual interval-lift argument controls the singularity stabilizers. An (n−1)-connected band in the germ complex then gives type F_n for the simple derived subgroup. A Lipschitz retraction onto a kernel of endpoint characters retains the homological finiteness obstruction.

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• 10.5281/zenodo.234376442026

  Spectral criteria for extending partial left orders

  Martino Lupini · Orderability, Groups

  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.

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• 10.5281/zenodo.234200872026

  Bi-orderable soluble groups with only Conradian left orders

  Martino Lupini · Orderability, Groups

  We prove that a soluble bi-orderable group all of whose left orders are Conradian is locally nilpotent, without any finite-generation or countability assumption. Consequently, every subgroup of such a group also has only Conradian left orders. This answers the soluble, bi-orderable case of the subgroup question posed by A. H. Rhemtulla and recorded in 2009.

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• 10.5281/zenodo.234199962026

  Thompson's group F′ is an O*-group: Positive conjugacy semigroups in Thompson's group F′

  Martino Lupini · Thompson groups, Orderability, Groups

  We give an explicit countable simple example for the Mal’cev–Fuchs question whether a simple O*-group exists, posed in 1963 and 1965. Every bi-invariant partial order on the commutator subgroup F′ of Thompson’s group F extends to a bi-invariant total order. Every partial positive cone lies in one of four cones determined by slopes at the leftmost or rightmost support endpoint. The central ingredient is a factorization theorem for right-moving one-bump maps with the same support: equality of their two endpoint slopes is equivalent to a sufficiently large power of one being a product of the corresponding number of conjugates of the other, supported on that interval. The proof combines bounded drift in orbit coordinates with a uniform correction bound from the commutator width of F′.

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• 10.5281/zenodo.232847082026

  A three-generated abelian lattice-ordered group without an existentially closed prime extension

  Martino Lupini · Orderability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.231768272026

  Convex normal subgroups of ℓ-simple lattice-ordered groups

  Martino Lupini · Orderability, Groups

  We construct a countable nonabelian normal-valued lattice-ordered group G which is ℓ-simple and has a nontrivial proper ordinary convex normal subgroup N. Both N and its abelian lattice hull H are free abelian of countably infinite rank. The subgroup N is an antichain normally generated by one element, while H is proper, normal, cofinal and order dense. This answers Problem 1.1 of Bludov, Glass, Kopytov and Medvedev negatively. Finite triangular coefficient tables on lexicographically ordered blocks preserve the antichain. A positive element forces an ordinary convex normal subgroup to contain the entire next finite-table group; iteration therefore gives simplicity. A conjugating extension and order density show that every positive nonidentity element of G has a cofinal conjugacy class. For an ℓ-simple group, a strong order unit guarantees an affirmative answer. We derive strong order units from finite lattice generation, a finite-index commensurator of a nontrivial finitely generated subgroup, or a nontrivial normal subgroup whose elements have powers in a fixed finitely generated subgroup.

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• 10.5281/zenodo.231257522026

  The Artin group of type $E_8$ and the left-orderability of spherical Artin groups

  Martino Lupini · Orderability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.230466982026

  A countable lattice-ordered group not latticeisomorphic to any abelian one

  Martino Lupini · Orderability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.229325952026

  Lattice-Ordered Groups as Full Automorphism Groups of Posets

  Martino Lupini · Orderability, Posets, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.229324302026

  A subdirect product of Ohkuma groups with nonunique addition

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228833532026

  Isomorphism of finitely generated bi-orderable elementary amenable groups is universal

  Martino Lupini · Orderability, Amenability, Groups

  No abstract on this deposit.

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