Coxeter Groups and Abstract Elementary Classes : The Right-Angled Case

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Hyttinen , T & Paolini , G 2019 , ' Coxeter Groups and Abstract Elementary Classes : The Right-Angled Case ' , Notre Dame Journal of Formal Logic , vol. 60 , no. 4 , pp. 707-731 . https://doi.org/10.1215/00294527-2019-0027

Title: Coxeter Groups and Abstract Elementary Classes : The Right-Angled Case
Author: Hyttinen, Tapani; Paolini, Gianluca
Other contributor: University of Helsinki, Department of Mathematics and Statistics
Date: 2019-11
Language: eng
Number of pages: 25
Belongs to series: Notre Dame Journal of Formal Logic
ISSN: 0029-4527
DOI: https://doi.org/10.1215/00294527-2019-0027
URI: http://hdl.handle.net/10138/307407
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 L-omega 1,L- omega-sentence. We also observe that the isomorphism relation of any countable complete first-order theory is kappa-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.
Subject: classification theory
abstract elementary classes
Coxeter groups
AUTOMORPHISMS
RIGIDITY
111 Mathematics
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