Subject / 9 records

Geometric group theory

Quasi-isometry, coarse geometry, fixed points, finiteness properties, and profinite constructions.

Classification: Second Brain

9 records

• 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.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.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.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.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.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.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.228398652026

  UCT-universal nuclear quotients of a binary wreath product

  Martino Lupini · Mathematics

  No abstract on this deposit.

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