Determine whether hyperbolic non-crystallographic groups embed into crystallographic partners

Determine whether any known non-crystallographic hyperbolic Coxeter group can be embedded as a subgroup of a higher-rank crystallographic hyperbolic Coxeter group of the same general type, thereby forming a hyperbolic Coxeter pair.

Background

Euclidean Coxeter pairs underlying familiar quasicrystals are known, but the corresponding Lorentzian or hyperbolic classification is incomplete. The thesis notes that several non-crystallographic hyperbolic graphs are known, while no suitable higher-rank crystallographic partners had been identified for them at the point of writing.

This unresolved embedding problem limits the construction of spacetime sCNP quasicrystals with non-crystallographic symmetry groups beyond the infinite dihedral examples.