Around first-order rigidity of Coxeter groups

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.