Homogeneity in Coxeter groups and split crystallographic groups

We prove that affine Coxeter groups, even hyperbolic Coxeter groups and one-ended hyperbolic Coxeter groups are homogeneous in the sense of model theory. More generally, we prove that many (Gromov) hyperbolic groups generated by torsion elements are homogeneous. In contrast, we construct split crystallographic groups that are not homogeneous, and hyperbolic (in fact, virtually free) Coxeter groups that are not homogeneous (or, to be more precise, not $\mathrm{EAE}$-homogeneous). We also prove that, on the other hand, irreducible split crystallographic groups and torsion-generated hyperbolic groups are almost homogeneous. Along the way, we give a new proof that affine Coxeter groups are profinitely rigid. We also introduce the notion of profinite homogeneity and prove that finitely generated abelian-by-finite groups are profinitely homogeneous if and only if they are homogenenous, thus deducing in particular that affine Coxeter groups are profinitely homogeneous, a result of independent interest in the profinite context.