Subject / 40 records

Polish groups and descriptive complexity

Polishability, Borel complexity, orbit relations, and topological group invariants.

Classification: Second Brain

40 records

• 10.5281/zenodo.236824122026

  The noncommutative weak extension principle under OCA

  Martino Lupini · C*-algebras, Descriptive set theory

  We prove that the Open Colouring Axiom implies the noncommutative weak extension principle. Every *-homomorphism between coronas of separable nonunital C*-algebras therefore splits, by a commuting projection, into a part with Borel graph and a part with nonmeagre kernel. This removes Martin's Axiom from Vignati and Yilmaz's theorem and answers their proposed weakening for Theorem A. In particular, all such homomorphisms with stable source are Borel under the Open Colouring Axiom alone. We also obtain Borel product lifts for bounded linear maps between reduced products of separable Banach spaces and separable C*-algebras which respect the canonical coordinate restrictions.

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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.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.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.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.234044292026

  Spectral ranks and dynamical lengths of preliminary C*-algebras

  Martino Lupini · C*-algebras, Descriptive set theory

  We classify the spectral ranks and automorphism lengths of nonzero separable preliminary C*-algebras, whose irreducible representations are all finite-dimensional. The spectral invariants are the Fell rank ρ and the Pedersen rank β of the irreducible-dimension function. Automorphism length q measures the transfinite descriptive complexity of automorphisms modulo those implemented by multiplier unitaries; its fractional values distinguish terminal Borel classes.

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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.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.233421812026

  Bounded geometry in Polish groups: locally compact models, amenability, and simple examples

  Martino Lupini · Orderability, Descriptive set theory, Amenability

  We develop locally compact models and applications for Polish groups of bounded geometry. Building on Hrushovski's quasi-model theorem and the ultrafilter framework of Krupiński–Pillay, we give an explicit coarse-density calculation for a model with compact conjugation-invariant multiplication defect and exact control of the intrinsic coarsely bounded bornology. Applications yield the Haagerup property under topological amenability and two-sided entropy estimates under metric amenability. Packing and model obstructions treat the stated classes of groups. For McCleary's classical simple periodic coloured-order group, we compute the pointwise Polish topology and intrinsic coarse geometry, obtaining a non-locally compact quasi-line.

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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.233411642026

  Solution to Dixmier's Problem about spectra of C*-algebras

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

  Dixmier asked in 1967 whether the spectra of simple separable non-type-I C*-algebras all have the same Borel structure. We solve the formulation using ordinary isomorphisms of quotient Borel σ-algebras by proving that its universal assertion is independent of ZFC, relative to the consistency of ZFC. Under the continuum hypothesis, all separable non-type-I spectra are measurably isomorphic, without simplicity or unitality assumptions. In the model obtained by iterating Sacks forcing for ω₂ stages with countable supports over a GCH ground model, a fixed simple unital universal spectrum is not measurably isomorphic to the CAR spectrum. Neither consistency direction requires a large cardinal.

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• 10.5281/zenodo.232914592026

  Sharp Borel representation dimensions for compact arithmetic actions

  Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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• 10.5281/zenodo.232914432026

  Arithmetic denominators and ordinary compact orbit comparisons

  Martino Lupini · Mathematics

  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.232767792026

  Local fields and ordinary affine Borel comparisons

  Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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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.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.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.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.228829562026

  Rank-two and two-prime rigidity in Borel classification problems

  Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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• 10.5281/zenodo.228433542026

  Nonabelian extensions of Solecki's coset dichotomy

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.228294622026

  LoCoLoCo Polish subgroups and the non-Archimedean complexity spectrum

  Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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• 10.5281/zenodo.228167792026

  Joint Euler-cocycle estimates and simplicial volume of surface bundles

  Martino Lupini · Mathematics

  No abstract on this deposit.

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• 10.5281/zenodo.228133152026

  Malliavin--Malliavin and whirly amenability: Borel observations, translation displacement, and total variation

  Martino Lupini · Descriptive set theory, Amenability

  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.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.227011632026

  Borel-definable homological algebra for locally compact Polish modules over rings of integers

  Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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• 10.5281/zenodo.227010812026

  Homological algebra of pro-Lie Polish modules over rings of integers

  Martino Lupini · Descriptive set theory

  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.226861732026

  Corrigendum to "Complexity Classes of Polishable Subgroups"

  Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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• 10.5281/zenodo.226833362026

  On the Borel complexity of operator images

  Martino Lupini · Operator algebras, Descriptive set theory

  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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• 10.5281/zenodo.223388852026

  The Descriptive Complexity of Nonabelian $\boldsymbol{\lim^1}$

  Matteo Casarosa, Martino Lupini · Descriptive set theory

  No abstract on this deposit.

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