Coset intersection complexes for graph products of finite groups

Determine the homotopy type of the coset intersection complex associated to maximal large join subgroups of a graph product of finite groups.

Background

The paper’s homotopy-type theorem for coset intersection complexes applies to graph products of infinite groups. The authors seek an analogue for graph products of finite groups, including right-angled Coxeter groups, both to obtain commensurability invariants and to enable comparisons with right-angled Artin groups. A large join is defined as a join whose factors each contain at least two non-adjacent vertices.

References

Let \Gamma be a finite graph and \mathcal{G} a collection of finite groups indexed by \Gamma. What is the homotopy type of the coset intersection complex $$\mathcal{K} \left( \Gamma \mathcal{G}, { \langle \Lambda \rangle, \Lambda \subset \Gamma \text{ maximal large join} } \right)?$$

— 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 “Right-angled Coxeter groups,” Question 6.1