Quasi-isometric preservation of maximal join cosets in graph products

Establish whether every quasi-isometry between graph products of finitely generated infinite groups sends each coset of a maximal join subgroup within uniformly bounded Hausdorff distance of a coset of a maximal join subgroup in the target graph product.

Background

For right-angled Artin groups, the paper proves that maximal join subgroups are coarsely preserved by quasi-isometries. The authors ask whether the analogous statement holds for graph products of arbitrary finitely generated infinite groups. A positive answer would extend the quasi-isometric invariant obtained for right-angled Artin groups to more general graph products.

References

Given a quasi-isometry \eta : \Gamma_1 \mathcal{G}_1 \to \Gamma_2 \mathcal{G}_2, does there exist a constant C \geq 0 such that \eta sends every coset of a maximal join subgroup of \Gamma_1 \mathcal{G}_1 at Hausdorff distance \leq C from the coset of a maximal join subgroup of \Gamma_2 \mathcal{G}_2?

— Homotopy types of complexes of hyperplanes in quasi-median graphs and applications to right-angled Artin groups  (2503.08411 - Abbott et al., 11 Mar 2025) in Section 6, paragraph “Other quasi-isometric invariants,” Question