Description of the monolith of the type-preserving tree-lattice commensurator

Determine whether the monolith M_d of C_d is equal to the subgroup of the type-preserving subgroup C_d^+ generated by all subgroups of C_d^+ that are commensurable with W ∩ C_d^+, where W is the standard free Coxeter cocompact lattice in Aut(T_d).

Background

The paper recalls that M_d contains W ∩ C_d+ and considers the normal subgroup generated by subgroups of C_d+ commensurable with this lattice subgroup. This generated subgroup is known to contain M_d, but it is not known whether the containment is an equality; a positive answer would parallel the corresponding result for the abstract commensurator of a free group.

References

Is the monolith $M_d$ equal to the subgroup of $C_d+$ generated by subgroups of $C_d+$ that are commensurable with $W \cap C_d+$ ?

— Free abelian quotients of commensurators  (2609.35323 - Boudec, 28 Sep 2026) in Section 5, subsection “Further questions”