Compact special cube complexes for Coxeter groups

Determine whether every Coxeter group admits a compact special cube complex whose fundamental group is a finite-index subgroup of the Coxeter group.

Background

The paper notes that Coxeter groups are virtually fundamental groups of special cube complexes, a fact used to distinguish the paper’s finite-generation result for automorphism groups from earlier results concerning compact special groups. The unresolved issue is whether the special cube complexes in this virtual-speciality statement can always be chosen to be compact. A positive answer would potentially enable techniques developed for compact special groups to apply more directly to Coxeter groups, while the paper emphasizes that this question is not resolved by its methods.