Coxeter groups and abstract elementary classes: the right-angled case. (English) Zbl 1472.03032

Hyttinen, Tapani; Paolini, Gianluca

Notre Dame J. Formal Logic 60, No. 4, 707-731 (2019).

Summary: 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 \(L_{\omega_1,\omega}\)-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.

MSC:

03C48	Abstract elementary classes and related topics
03C60	Model-theoretic algebra
20F55	Reflection and Coxeter groups (group-theoretic aspects)

Keywords:

classification theory; abstract elementary classes; Coxeter groups

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