Around first-order rigidity of Coxeter groups

Abstract

By the work of Sela, for any free group $F$, the Coxeter group $W_3 = \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z} \ast \mathbb{Z}/2\mathbb{Z}$ is elementarily equivalent to $W_3 \ast F$, and so Coxeter groups are not closed under elementary equivalence among finitely generated groups. We study what happens after restricting to models generated by finitely many torsion elements, which we call finitely torsion-generated. We prove that if $(W,S)$ is a Coxeter system whose irreducible components are finite, affine, or non-elementary hyperbolic, and $G$ is finitely torsion-generated and elementarily equivalent to $W$, then $G$ is a Coxeter group. This combines results from [22, 25] with the following hyperbolic result: if $W$ is a hyperbolic Coxeter group and $G$ is finitely torsion-generated and $\mathrm{AE}$-equivalent to $W$, then $G$ is isomorphic to one of finitely many Coxeter groups. We also construct two non-isomorphic hyperbolic Coxeter groups that are $\mathrm{AE}$-equivalent. We then consider first-order torsion-rigidity, meaning that $W$ is the only finitely torsion-generated model of its theory, and prove it for even hyperbolic Coxeter groups and for free products of one-ended or finite hyperbolic Coxeter groups. We conjecture that analogous phenomena hold for all Coxeter groups. Finally, we prove that elementarily equivalent even Coxeter groups are isomorphic, generalizing the analogous result for right-angled Coxeter groups from [11].