Around first-order rigidity of Coxeter groups

TL;DR

This work addresses when a finitely generated Coxeter group W is determined up to isomorphism by its first-order theory, focusing on models G generated by finitely many torsion elements. The authors develop a robust framework combining JSJ/centered splittings, tree-of-cylinders methods, preretractions, and domain arguments to prove that AE-equivalence with a Coxeter W forces G to be a Coxeter group in broad cases (spherical/affine/hyperbolic irreducibles), with finiteness statements for hyperbolic cases and instances of first-order torsion-rigidity. They also exhibit AE-equivalent but non-isomorphic hyperbolic Coxeter groups, illustrating limits of rigidity, and prove that for even Coxeter groups elementary equivalence implies isomorphism, establishing a strong rigidity phenomenon in this subclass. The results blend geometric group theory and model theory, leveraging finite-by-orbifold structures, Stallings/JSJ theory, and the tree-of-cylinders to tightly constrain possible AE-equivalent models. Overall, the paper advances the understanding that, under torsion-generated constraints, being a Coxeter group can be seen as a first-order property in substantial parts of the Coxeter landscape, with wide-reaching implications for elementary theory and profinite rigidity in 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. In this paper we show that if we restrict to models which are generated by finitely many torsion elements (finitely torsion-generated), then we can recover striking rigidity results. Our main result is that if $(W, S)$ is a Coxeter system whose irreducible components are either spherical, or affine or (Gromov) hyperbolic, and $G$ is finitely torsion-generated and elementarily equivalent to $W$, then $G$ is itself a Coxeter group. This combines results of the second author et al. from [MPS22, PS23] with the following main hyperbolic result: if $W$ is a Coxeter hyperbolic group and $G$ is $\mathrm{AE}$-equivalent to $W$ and finitely torsion-generated, then $G$ belongs to a finite collection of Coxeter groups (modulo isomorphism). Furthermore, we show that there are two hyperbolic Coxeter groups $W$ and $W'$ which are non-isomorphic but $\mathrm{AE}$-equivalent. We also show that, on other hand, if we restrict to certain specific classes of Coxeter groups then we can recover the strongest possible form of first-order rigidity, which we call first-order torsion-rigidity, namely the Coxeter group $W$ is the only finitely torsion-generated model of its theory. Crucially, we show that this form of rigidity holds for the following classes of Coxeter groups: even hyperbolic Coxeter groups and free products of one-ended or finite hyperbolic Coxeter groups. We conjecture that the same kind of phenomena occur for the whole class of Coxeter groups. In this direction, we prove that if $W$ and $W'$ are even Coxeter groups which are elementarily equivalent, then they are isomorphic.

Figures (4)

• Figure 1: The finite group $A$, isomorphic to $S_3\times S_7\times S_4$.
• Figure 2: The finite group $B$, isomorphic to $S_7\times S_4$.
• Figure 3: The finite group $C$, isomorphic to $(\mathbb{Z}/2\mathbb{Z})^5$.
• Figure :

Theorems & Definitions (110)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.11
• Proposition 2.1