Catalog

Repositories

Latest version of each Zenodo concept by Logica Artificialis. A row shows only the files on that record, and a GitHub address only when the manuscript names one.

71 repositories

• 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.238276522026

  Failure of rational injectivity for minimal principal étale groupoids

  Martino Lupini · Orderability, Groupoids, Groups

  Applying OpenAI's torsion-free group counterexample, we construct a second-countable locally compact Hausdorff minimal principal ample groupoid with Cantor unit space whose ordinary reduced Baum--Connes assembly map has an infinite-order kernel class in degree zero. Its rationalization is therefore not injective, with the natural coefficient algebra of continuous functions on the unit space. The groupoid is a transformation groupoid of a finitely generated torsion-free group and is compactly generated. We use the invariant-measure trace class of Antonini, Azzali and Skandalis to preserve an ordinary assembly kernel class under extension of coefficients to continuous functions. Elek's free minimal measure-preserving Cantor action supplies the required dynamics. The standard compact-action comparison identifies group assembly with coefficients and transformation-groupoid assembly. Thus minimality and principality do not restore the natural-coefficient reduced Baum--Connes assertion.

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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.237317872026

  A finitely presented nonindicable subgroup of a free profinite group

  Martino Lupini · Groups

  We construct an infinite, two-generated, finitely presented subgroup H of the free profinite group of rank two whose abelianization is (ℤ/60ℤ)^2. Thus free profinite groups of rank at least two fail to be locally indicable, disproving a conjecture of Jaikin-Zapirain. The group H has a normal subgroup of index 60 isomorphic to F_29 × F_19 × F_11, where F_r denotes the abstract free group of rank r. More generally, every nontrivial finite perfect group is a quotient of a finitely presented torsion-free group with finite abelianization that embeds in the free profinite group of rank two. The construction combines free products of finite cyclic groups over a common finite quotient, completing a different free kernel at each prime.

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• 10.5281/zenodo.236826802026

  Jordan spectra of nilpotent derivations from scalar geometry

  Martino Lupini · Orderability, Groups

  We determine the complete Jordan spectrum of nilpotent derivations of a nonabelian class-two Lie algebra I over a characteristic-zero field. We assume that its scalar bracket space has a decomposable basis over the ground field and that its reduced geometric decomposable locus admits no nontrivial additive-group action. A plane-preserver algebra and a hyperplane arrangement determine the horizontal rank set R; an integer criterion then decides every full Jordan type. Every admitted type occurs over the ground field, and the spectrum remains unchanged under field extension. For u = dim(I/Z(I)), d = dim I′, f = dim(Z(I)/I′), and r = max R, the greatest partition is (2^r, 1^(u-2r)) + (f) + (1^d), with componentwise addition. The full spectrum recovers R and is a dominance order ideal exactly when R has no gaps. For fixed free central dimension f, finite scalar loci with identical support-sum rank functions can have different spectra; support dimension six is minimal. A family has Jordan strata and commuting varieties whose component dimensions differ without bound. Geometric consequences include a smooth rational greatest-type stratum, a crepant resolution, and a classification of nilpotent linear spaces of largest dimension. Hall polynomials give uniform Jordan-stratum point counts at almost all places for number-field models.

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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.234699552026

  Order, norming faces, and symmetries of Tursi's Banach lattice

  Martino Lupini · Orderability, Groups

  Let L be Tursi’s separable universal homogeneous real Banach lattice. The set of decreasing positive sequences with infimum zero is coanalytic complete, even for sequences of components of one fixed quasi-interior vector, providing a fixed-lattice coanalytic-complete example for the decreasing-order-complexity question raised in Tursi’s 2024 survey. This gives a negative answer to the 2024 question of Avilés, Rosendal, Taylor, and Tradacete whether every separable Banach lattice belongs to their countable Fatou hierarchy. Unrestricted sequential order convergence has the exact complete complexity coA(Pi^1_1), the dual of one Souslin operation on coanalytic sets. We classify quasi-interior positive vectors by their norming Choquet simplices and answer Tursi’s documented 2021 automorphism-group universality question affirmatively. We prove universality of embedding orbits for every nonzero separable domain and of automorphism conjugacy, and establish concentration, lifting, and representation results for pointwise face kernels. Further results describe generic embedding orbits, ideals, and finite-dimensional approximate disposition.

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• 10.5281/zenodo.234611402026

  Endomorphism counts from primitive orbits and synchronized block actions

  Martino Lupini · Orderability, Posets, Groups

  For a finite poset P, let R(P) = |End(P)|/|Aut(P)|, where endomorphisms preserve the reflexive order. We prove R(P) ≥ 2^{|P|/(64 log₂ |P|)} for |P| ≥ 4 when every chain has at most two elements and the action on each nonsingleton automorphism orbit is primitive. The same bound holds for a class of imprimitive actions: on each interaction component the action must be the full automorphism group of a rooted tree, acting identically on all the identified orbits. The proof combines simultaneous collapses of indices with maps into principal ideals and filters. We give counting lemmas that retain the whole kernel of a block action, and a quantitative collapse, degree, or orbit-size alternative for normal products of alternating groups with natural orbits. We also construct connected twin-free posets with group S₃ wr Sₜ, constant minimum of the two maximum principal-set sizes, and imprimitive action on every orbit, although every orbit-preserving endomorphism is an automorphism. A restricted three-level extension has lower bound 2^{|P|/(128 (log₂ |P|)²)}. These results give partial answers to the finite-poset automorphism problem of Rival and Rutkowski; the unrestricted question remains beyond the arguments here.

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• 10.5281/zenodo.234527882026

  The small index property for the random poset

  Martino Lupini · Posets, Descriptive set theory, Groups

  Let P be the countable universal homogeneous poset. We prove that every abstract subgroup H ≤ Aut(P) with [Aut(P):H] < 2^{ℵ₀} contains the pointwise stabilizer of a finite subset of P. Thus Aut(P) has the small index property, answering the question recorded as open by Barbina and Macpherson in 2005. No regularity assumption on H is required. The proof equips P with a generic strictly increasing rational-valued rank. Its stabilizer has ample generics, and generic pairs of ranks yield a comeager double coset. We show that the small index property passes from a closed subgroup with such a double coset to the ambient Polish group.

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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.234301632026

  Completely positive coordinates for KK-theory and relative uniform stability

  Martino Lupini · C*-algebras, Amenability, Groups

  Let A be a unital nuclear C*-algebra, M a von Neumann algebra, and I a norm-closed ideal of M. We prove that a unital completely positive map Φ:A→M, multiplicative modulo I, admits a unital homomorphic correction with the same quotient map whenever

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• 10.5281/zenodo.234249602026

  Robust fixed points and finite classifying spaces: CAT(0) actions, Banach targets and group fillings

  Martino Lupini · Groups

  For a finitely presented group, we characterize the fixed-point property for all isometric actions on complete CAT(0) spaces by finite systems of polynomial distance inequalities. An infeasible system gives a uniform tolerance for simultaneous errors in the generating maps, defining relations and midpoint geometry. For finite presentations, this yields an algorithm that halts exactly when the property holds. On genuine CAT(0) spaces and prescribed Banach classes, continuous iterations give locally Hölder retractions onto common fixed sets and continuous choices in parameter families. For one arithmetic presentation we evaluate a nonlinear generator tolerance of 10−232 on genuine CAT(0) spaces.

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• 10.5281/zenodo.234220762026

  Isomorphism is strictly Borel reducible to quasi-isometry for finitely generated groups

  Martino Lupini · Descriptive set theory, Groups

  We prove that isomorphism of finitely generated groups is strictly Borel reducible to quasi-isometry, confirming a conjecture of Simon Thomas. More precisely, we construct continuous reductions to quasi-isometry of both the universal countable Borel equivalence relation and the relation E1 of eventual equality of sequences of reals. Thus quasi-isometry of finitely generated groups is not Borel reducible to the orbit relation of any Borel action of a Polish group. In particular, it is not essentially countable, answering a further question of Thomas. The constructions use doubles of a fixed finitely generated group over infinitely generated subgroups. Geodesic patterns recover the subgroups up to commensurability after conjugation in a fixed pattern-symmetry group, while an explicit gluing argument makes balanced finite-index changes invisible to quasi-isometry.

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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.234182732026

  Isomorphism and bi-embeddability of countable Archimedean ordered groups

  Martino Lupini · Orderability, Descriptive set theory, Groups

  Let E and F denote order-preserving isomorphism and mutual order-preserving embeddability of countable Archimedean ordered groups. We prove that E

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• 10.5281/zenodo.234055112026

  Bounded coordinate lengths and normal closures in Polish products

  Martino Lupini · Descriptive set theory, Groups

  We characterize when the abstract normal closure of an element of a countable product of arbitrary Polish groups admits a Polish group topology with the inherited Borel sets. For h=(hᵢ) in ∏ᵢLᵢ, put Bᵢ=hᵢ^{Lᵢ}(hᵢ⁻¹)^{Lᵢ} and Pᵢ=⟨Bᵢ⟩. The normal closure is Polishable precisely when every Pᵢ is Polishable and, for some integer k, Pᵢ=Bᵢᵏ outside finitely many coordinates. In this case it is the countable cyclic extension (∏ᵢPᵢ)⟨h⟩. Failure of the uniform bound yields a continuous injective reduction of E₁, eventual equality of sequences in 2^ℕ, to the coset relations of both the balanced and ordinary normal closures. The argument applies more generally to symmetric analytic coordinate generating sets. We also give an exact criterion for subgroups defined by uniformly bounded finite Borel coordinate lengths; their Polishable branch has coset relation either trivial or Borel bireducible with E₀. Applications include arbitrary nonabelian abstractly simple Polish factors and compact connected Lie factors. For projective unitary factors, a uniform spectral condition is equivalent to Polishability. Examples distinguish the coordinate obstruction, the uniform-width obstruction, and the cyclic obstruction to closedness.

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• 10.5281/zenodo.233963312026

  Support rings, Polishability, and the SIN property

  Martino Lupini · Descriptive set theory, Groups

  Let T be a finite nontrivial group and I countable. We prove that a Borel subgroup P ≤ TI is Polishable whenever its realized supports form a Polishable Boolean ring ℬ and each principal-support group PA, A ∈ ℬ, has an exact coordinate frame over T[ℬ|A]. This includes groups locally contained in finite direct powers of a finite quasisimple group. In that case, P admits a compatible Polish SIN topology exactly when its support ring admits a compatible Polish additive topology with a downward-closed neighborhood base. Solid support topologies also give a global frame and a complete bi-invariant metric. For abstract groups locally contained in finite powers of a finite nonabelian simple group S, the same SIN criterion uses the Boolean ring of principal normal subgroups; its locally compact instances are SJ × D, with J at most countable and D countable discrete. We further prove that principally trivial Boolean-ring covers preserve Polishability for every finite deck group. The proofs use frame-independent category and Polish group extensions. Explicit counterexamples delimit the coefficient, coordinate-projection, intrinsic and finite-deck hypotheses.

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• 10.5281/zenodo.233835602026

  Proximally irreducible affine extensions: composition and maximal covers

  Martino Lupini · Groups

  We prove that the composition of proximally irreducible affine extensions is proximally irreducible, answering a question of Bal for arbitrary topological groups and compact Hausdorff affine flows. The proof combines localization, Baire category, and compactness for finite convex witnesses, without requiring closed extreme-point sets. Composition, factorization, inverse limits, and uniform density bounds yield maximal proximally irreducible covers, unique up to a unique isomorphism over the base and characterized by a universal property. Extensions of a fixed base form a complete lattice, realized by closed invariant spaces of affine functions on its maximal cover. Passing to probability flows recovers maximal proximal covers of minimal flows and gives canonical refinements, pullbacks, and functorial lifts. For the trivial group, Gleason covers describe maximal domains and classify polytope extension lattices. For separable groups, extensions of metrizable bases admit metrizable approximations that detect nontriviality. Examples delimit the affine pullback and metrizability statements.

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• 10.5281/zenodo.233625992026

  Exactness and amenability at infinity for locally compact groupoids

  Martino Lupini · Groupoids, Amenability, Groups

  For a second-countable locally compact Hausdorff étale groupoid G, we prove the equivalence of exactness of Cr*(G), exactness of reduced crossed products, and strong amenability at infinity. These conditions are also equivalent to exactness of any one twisted reduced algebra. The amenable action can be chosen with a proper open surjective anchor admitting a continuous section. This answers Anantharaman–Delaroche’s 2016 exactness questions and the étale case of LaLonde’s 2017 question about exact transformation groupoids. Consequently, exactness together with canonical weak containment implies amenability; for compact unit space, exactness is equivalent to amenability of the Furstenberg boundary action. We also prove permanence of strong amenability at infinity under continuous strongly surjective homomorphisms between second-countable groupoids with Haar systems, when the kernel and target have this property. No Haar system on the kernel is required. Finally, we construct an inner-exact étale group bundle with exact fibers and nonexact reduced algebra, supplying the example asked for by Anantharaman–Delaroche in 2016 and answering LaLonde’s 2017 group-bundle question negatively. Its fibers are hyperbolic groups converging strongly to a free group, while their Cayley graphs fail uniform property A.

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• 10.5281/zenodo.233440132026

  Injective and projective objects in categories of topological torsion Polish abelian groups

  Martino Lupini · Descriptive set theory, Groups

  We give complete proofs of the injective and projective classifications in four categories of non-Archimedean Polish abelian groups, according to whether the groups are locally compact and whether their discrete quotients are torsion or p-primary. The new results characterize projectives without a local compactness assumption: they are precisely the compact torsion-free groups, namely products over primes p of powers of Z_p in the torsion case and powers of Z_p in the p-primary case, with all multiplicities at most countable. These results answer two questions posed by Casarosa, Codenotti, and Lupini in 2024. We also prove their known injective classifications: the locally compact injectives are the countable discrete divisible groups of the relevant prime class, and the injectives in the larger categories are countable products of such groups. None of the four categories has enough projectives, and precisely the compact objects admit strict projective resolutions. The two larger categories have enough injectives; in the locally compact categories, precisely the discrete objects embed strictly into injectives. We also characterize injectives and projectives in all four left hearts: only zero is injective, and the projectives remain exactly the compact torsion-free groups. The locally compact left-heart results are known; for the two larger hearts, the absence of nonzero injectives follows from an argument using Solecki ranks.

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• 10.5281/zenodo.233421992026

  Efficient contractions of the Urysohn isometry group

  Martino Lupini · Descriptive set theory, Groups

  We construct a contraction of the Urysohn isometry group with uniform and coarse control. Pointwise stabilizers of nonempty finite sets admit contractions uniformly continuous for the left uniformity, and the global contraction preserves bounded-displacement sets. Ordinary contractibility is due to Melleray, while the quantitative realization and embedding inputs are due to Lešnik. The construction here adds guided surjective back-and-forth, inverse evaluation control, and uniform endpoint estimates. A preliminary bounded-chain argument shows that the bounded-image clause of efficient contractibility follows from the stated uniform-continuity hypothesis.

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• 10.5281/zenodo.233421522026

  Pointwise bounded cocycles and coarse decompositions of topological groups

  Martino Lupini · Descriptive set theory, Groups

  We prove a uniform-boundedness principle for Borel cocycles between Polish groups: pointwise coarsely bounded fibres are uniformly coarsely bounded over each coarsely bounded subset of the acting group. A bounded-defect version applies to quasicocycles. The principle yields a Zappa–Szép coarse-product criterion without local boundedness and a nonsplit-extension criterion using Borel sections and pointwise intrinsic bounds, without a centralizing-section hypothesis. We also prove that local boundedness lifts through ultralocally bounded quotients. Supplementary sections give conditional coupling-composition results and examine category assumptions in factorization, using a classical additive-graph construction.

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• 10.5281/zenodo.233421262026

  Boundedness exponents of unitary function groups and measured full groups

  Martino Lupini · Descriptive set theory, Groups

  Let $X$ be an infinite compact metrizable space and let $M\ne0$ be a von Neumann algebra with separable predual. For the strong topology on $\mathcal{U}(M)$, we classify boundedness of $C(X,\mathcal{U}(M))$: finite exponents occur exactly when $M$ is properly infinite, with least open-exhaustion exponent five and least finite-neighbourhood exponent three, and ordinary Property $(\mathrm{OB})$ holds exactly when $\dim X=0$ or $M$ has no matrix direct summand. Continuous fixed-involution factors give the finite-exponent result; compact-uniform adaptations of spectral regularization and central splitting give ordinary boundedness. The classification concerns function-group boundedness, while the corresponding target contractibility criterion is due to Jekel. Related four-factor target-unitary and type III full-group bounds are upper bounds.

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• 10.5281/zenodo.233421102026

  Exact boundedness exponents for permutation and automorphism groups

  Martino Lupini · Descriptive set theory, Groups

  We determine exact boundedness exponents for automorphism groups in the pointwise topology. For countably infinite-dimensional vector spaces over countable fields, general linear groups have least open-exhaustion exponent three over finite fields and four over infinite fields. Symplectic groups for the standard alternating form have corresponding exponents three and five. Orthogonal groups for the form $B(x,y)=\sum_i x_i y_i$ have the same three-versus-five dichotomy in characteristic different from two. The automorphism group of a free abelian group of countably infinite rank has least open-exhaustion exponent five. Routing and reflection give exact exponent-four results for unrestricted permutation wreath products. A directional fresh-copy criterion and function-group applications supplement these calculations.

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• 10.5281/zenodo.233420552026

  Quantitative boundedness in homeomorphism groups

  Martino Lupini · Descriptive set theory, Groups

  We obtain sharp boundedness exponents from finite-correction obstructions and continuously varying geometric normal forms. With the compact-open topology, the homeomorphism group of the two-sphere and the full and orientation-preserving homeomorphism groups of planar telescoping surfaces have least open-exhaustion exponent five and least finite-neighbourhood exponent three. The same pair holds for continuous-function groups from nonempty nondiscrete locally compact second-countable Hausdorff spaces into the boundary-fixing homeomorphism group of a ball of any positive dimension. Five-factor bounds for further geometric targets refine known qualitative boundedness arguments; the full-sphere bound and the pointwise boundary-fixing ball word already occur in published proofs. A controlled Hilbert-cube argument supplies a quantitative bound and repairs the earlier qualitative proof. We also extend ultralocal boundedness from closed compact manifolds to compact manifolds with boundary.

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• 10.5281/zenodo.233420182026

  Fixed words, function groups, and exact odd boundedness exponents

  Martino Lupini · Descriptive set theory, Groups

  For an infinite zero-dimensional compact metrizable space $X$ and a Polish group $P$, we characterize boundedness with a fixed number of factors of $C(X,P)$ by words alternating a target identity neighbourhood with fixed elements. If the least number of neighbourhood factors required for every identity neighbourhood of $P$ is $k$, then $C(X,P)$ has least finite-neighbourhood exponent $k$ and least open-exhaustion exponent $2k-1$. Neither exponent is finite when no such $k$ exists. Matching word bounds for circular covering groups then realize every odd least open-exhaustion exponent at least five, both among non-Archimedean Polish groups and among path-connected locally contractible Polish groups. We also establish an ordinary-boundedness criterion and transfer results for continuous and measurable function groups.

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• 10.5281/zenodo.233419432026

  Fixed-exponent boundedness and cocompact permanence in Polish groups

  Martino Lupini · Descriptive set theory, Groups

  We prove that closed cocompact inclusions and compact normal quotients preserve each fixed open-exhaustion and finite-neighbourhood exponent for Polish groups. The literal open-exhaustion conditions and the finite-neighbourhood conditions have distinct indices, related by an explicit interlacing calculus. We also obtain extension and product bounds. We also give a corrected proof of the previously stated coarse equivalence between a Polish group and a closed cocompact subgroup, using compact transition sets. The final section collects consequences and examples for groups whose coarsely bounded sets are relatively compact.

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• 10.5281/zenodo.233387882026

  Automatic equivariant Jiang–Su stability in the nonunital tracial setting

  Martino Lupini · C*-algebras, Amenability, Groups

  Let A be a separable simple nuclear C*-algebra that absorbs the Jiang–Su algebra Z and admits a nonzero densely defined lower semicontinuous trace. Using the Gaussian construction of the October 2026 OpenAI preprint, we prove that every action α of a countable discrete amenable group on A is cocycle conjugate to α ⊗ idZ. This answers the general tracial part of Szabó’s 2019 conjecture on automatic equivariant Jiang–Su stability, extending the October 2026 unital theorem to algebras with possibly unbounded traces and to trace-scaling actions. The extension applies that construction in a tracial quotient of Kirchberg’s central sequence algebra and uses the nonunital absorption results of Szabó and Wouters. The transfer is expressed as a local equivariant Gamma criterion for nonseparable unital algebras admitting relative copies of Z. Its uniform trace-splitting estimates allow the original action and its full cone of traces to be retained. Combined with the established purely infinite case, the result gives automatic equivariant Jiang–Su stability for all separable simple nuclear Z-stable algebras.

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• 10.5281/zenodo.232914082026

  Measurable signatures and algebraic structure of L¹ full groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.232914012026

  Local swaps, Polish group topologies, and automorphisms of L¹ full groups

  Martino Lupini · Descriptive set theory, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.232913922026

  Polish topologies and coset obstructions for measurable transformation groups

  Martino Lupini · Descriptive set theory, Groups

  No abstract on this deposit.

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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.231718172026

  Tame functions and Asplund representations of additive Banach groups

  Martino Lupini · Operator algebras, Groups

  We prove that tame functions on the additive group of c0 do not determine its norm topology, answering questions of Megrelishvili and Glasner–Megrelishvili. The proof combines Gowers’ positive-block stabilization with Megrelishvili’s independent-translates configuration. The obstruction extends by homomorphism pullback to Banach spaces containing c0. Full-sphere stabilization gives arbitrarily large subspace balls of approximate constancy and shows that every point of a closed norm ball lies in the tame closure of its boundary sphere. In contrast, a dyadic phase map and the coordinate-rotation construction of Antunes, Ferenczi, Grivaux and Rosendal give a strong-operator embedding of c0 into the isometry group of an Asplund space isomorphic to c0, with almost periodic scalar coefficients. Thus c0 is Asplund but not reflexively representable, settling the Asplund-versus-reflexive problem; even its tame algebra fails to determine the topology. Applications give local copies of βℕ in every proper semigroup compactification and exclude c0 from compact-open automorphism groups of finite-rank compact topological median algebras. For locally compact abelian H, tame functions determine the uniform topology of c0(H) exactly when H is totally disconnected. For second-countable H, these groups are always Asplund representable.

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• 10.5281/zenodo.231695762026

  Polishability from orbit reducibility of coset equivalence relations

  Martino Lupini · Descriptive set theory, Groups

  We prove that a Borel subgroup H of a Polish group G is Polishable if and only if its coset equivalence relation is Borel reducible to the orbit equivalence relation of a Borel action of a Polish group. The proof combines Jankov–von Neumann uniformization with an invariant category criterion derived from Solecki's approximation theorems. Averaging the canonical category ideals of the target orbits over G supplies the invariant ideal needed for recognition. Consequently, a non-Polishable Borel subgroup has a coset relation that is not classifiable by countable structures.

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• 10.5281/zenodo.231424712026

  The linear isometry group of the Holmes space

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.231332632026

  Universal minimal flows of automorphism groups of locally finite trees

  Martino Lupini · Groups

  No abstract on this deposit.

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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.231141372026

  Injective objects in the category of Polish abelian groups

  Alessandro Codenotti, Martino Lupini · Descriptive set theory, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.231060642026

  Constructive normal identities in metabelian groups

  Martino Lupini · 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.230461692026

  Property (OB) and the Coarse Geometry of Polish Groups --- Fixed Exponents, Permanence, and Applications

  Martino Lupini · Descriptive set theory, 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.228862952026

  Convergence relations of pro-Lie Polish groups

  Martino Lupini · Descriptive set theory, 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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• 10.5281/zenodo.228452362026

  Skew-amenability in finite-regularity topologies for path, loop, and gauge groups

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228399152026

  Charmenability of Leavitt unit groups and Clifford-lamp groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228340732026

  Contractibility and Banach-space representations of the Carderi--Thom groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228131892026

  Distortion of Banach-space unit spheres under their linear isometry groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.228097652026

  Generic monothetic subgroups of the measure automorphism group

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227764742026

  Universal minimal flows of homeomorphism groups of solenoids and Knaster continua

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227750142026

  Exactness of strong unitary groups associated with discrete groups with the approximation property

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227646152026

  Universal minimal flows of automorphism groups\\of stable Kirchberg algebras

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227582302026

  Borel invariant means on countable inverse limits of amenable groups

  Martino Lupini · Descriptive set theory, Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227493622026

  Whirly monothetic groups that are not L\'evy

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227436042026

  Amenability and skew-amenability of C1 path, loop, and one-dimensional gauge groups

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227378542026

  Concentrating measures and whirly amenability of Doucha's universal metric group

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227375692026

  Universal minimal flows of inductive limits of Lie groups

  Alessandro Codenotti, Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227224532026

  Extreme amenability of the automorphism group of the Jiang-Su algebra

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227221902026

  Extreme amenability of the automorphism group of the Razak-Jacelon algebra

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227189882026

  Universal minimal flows of Cuntz automorphism and unitary groups

  Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.227135582026

  Extreme amenability of the additive group of $L^p$ for $0<p<1$

  Martino Lupini · Amenability, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.226953502026

  Descriptive classification of classes of C*-algebras and locally compact groups

  Martino Lupini · C*-algebras, Descriptive set theory, Groups

  No abstract on this deposit.

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• 10.5281/zenodo.226898792026

  The franks of Moore groups

  Alessandro Codenotti, Martino Lupini, Francesco Milizia · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.226763992026

  On Niemiec's Universal Valued Groups

  Alessandro Codenotti, Martino Lupini · Groups

  No abstract on this deposit.

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• 10.5281/zenodo.225586172026

  Derived limits and phantom filtrations for towers of locally compact Polish abelian groups

  Martino Lupini · Descriptive set theory, Groups

  No abstract on this deposit.

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