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

TL;DR

This paper explores the model-theoretic properties of right-angled Coxeter groups, establishing criteria for their classification as abstract elementary classes and constructing examples with specific tameness and categoricity features.

Contribution

It introduces combinatorial criteria for classifying right-angled Coxeter groups as AECs, and constructs concrete examples demonstrating various model-theoretic properties.

Findings

The class of all right-angled Coxeter groups is not smooth.

Certain classes are shown to be tame and uncountably categorical.

A machinery is developed to build AECs with desired properties.

Abstract

We study classes of right-angled Coxeter groups with respect to the strong submodel relation of parabolic subgroup. We show that the class of all right-angled Coxeter group is not smooth, and establish some general combinatorial criteria for such classes to be abstract elementary classes, 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 translate into a machinery to build concrete examples of $\mathrm{AECs}$ satisfying given model-theoretic properties. We exhibit the power of our method constructing three concrete examples of finitary classes. We show that the first and third class are non-homogeneous, 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.