Identify hyperbolic Coxeter pairs with looped diagrams

Determine whether bicolourable hyperbolic Coxeter diagrams containing loops satisfy the eigenvalue-sharing condition required to form Coxeter pairs between non-crystallographic and crystallographic groups.

Background

The thesis studies whether non-crystallographic hyperbolic Coxeter groups can be embedded into higher-rank crystallographic groups through graph foldings. For tree-like simplicial graphs, an exhaustive computational search found no suitable pairs, although one candidate shared eigenvalues without yielding an explicit folding.

The authors identify bicolourable diagrams with loops as an unexamined class and propose testing their bicoloured Coxeter elements against the eigenvalue-sharing condition.