Finite-index right-angled subgroups of the Lee–Marquis examples

Determine whether any of the Lee–Marquis reflection-group examples admit finite-index subgroups that are right-angled reflection groups.

Background

Lee and Marquis constructed convex cocompact hyperbolic reflection groups in dimension five whose limit sets are 3-spheres but whose finite-index subgroups do not embed discretely in the isometry group of hyperbolic four-space. The paper emphasizes that these examples are not right-angled, unlike the examples constructed here, and explicitly leaves unresolved whether any of them possess finite-index right-angled reflection subgroups.

References

Previously, Lee--Marquis\S A.2 exhibited reflection groups as in Theorem~\ref{main} that are not right angled (nor does it seem clear whether any of their examples admit finite-index right-angled reflection subgroups), and indeed, their argument uses crucially that their polyhedra have mutually intersecting walls.

— Convex cocompact right-angled Gromov-Thurston polyhedra  (2608.19640 - Douba, 20 Aug 2026) in Introductory discussion following Theorem 1