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November 2019 Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case
Tapani Hyttinen, Gianluca Paolini
Notre Dame J. Formal Logic 60(4): 707-731 (November 2019). DOI: 10.1215/00294527-2019-0027

Abstract

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of a parabolic subgroup. We show that the class of all right-angled Coxeter groups is not smooth and establish some general combinatorial criteria for such classes to be abstract elementary classes (AECs), for them to be finitary, and for them to be tame. We further prove two combinatorial conditions ensuring the strong rigidity of a right-angled Coxeter group of arbitrary rank. The combination of these results translates into a machinery to build concrete examples of AECs satisfying given model-theoretic properties. We exhibit the power of our method by constructing three concrete examples of finitary classes. We show that the first and third classes are nonhomogeneous and that the last two are tame, uncountably categorical, and axiomatizable by a single Lω1,ω-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is κ-Borel reducible (in the sense of generalized descriptive set theory) to the isomorphism relation of the theory of right-angled Coxeter groups whose Coxeter graph is an infinite random graph.

Citation

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Tapani Hyttinen. Gianluca Paolini. "Coxeter Groups and Abstract Elementary Classes: The Right-Angled Case." Notre Dame J. Formal Logic 60 (4) 707 - 731, November 2019. https://doi.org/10.1215/00294527-2019-0027

Information

Received: 7 February 2017; Accepted: 19 August 2018; Published: November 2019
First available in Project Euclid: 14 September 2019

zbMATH: 07167765
MathSciNet: MR4019869
Digital Object Identifier: 10.1215/00294527-2019-0027

Subjects:
Primary: 03C48
Secondary: 05E15

Rights: Copyright © 2019 University of Notre Dame

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Vol.60 • No. 4 • November 2019
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