Discrete four-dimensional embeddings of the Lee–Marquis examples

Determine whether an argument analogous to the construction in this note can rule out discrete embeddings of the Lee–Marquis reflection-group examples into \(\mathrm{Isom}(\mathbb{H}^4)\).

Background

The paper’s main construction proves that certain right-angled reflection groups in Isom(H5)\mathrm{Isom}(\mathbb{H}^5) have no finite-index subgroups admitting discrete embeddings into Isom(H4)\mathrm{Isom}(\mathbb{H}^4). The authors note that the Lee–Marquis examples have many dihedral angles that are odd submultiples of π\pi, which prevents the same argument from applying straightforwardly, and explicitly state that it is unclear whether a similar argument can establish the corresponding non-embedding result for those examples.

References

By contrast, our approach is closer to that already suggested in the work of Gromov--Thurston\S3.7. As pointed out to the author by Gye-Seon Lee, the Lee--Marquis examples are also somehow in the spirit of Gromov--Thurston since they are obtained from Esselmann's compact $4$-dimensional hyperbolic polyhedra by deforming certain angles, though it seems unclear whether an argument similar to that contained in this note can rule out discrete embeddings of the Lee--Marquis examples into $\mathrm{Isom}(\mathbb{H}4)$, essentially because of the abundance of angles that are odd submultiples of $\pi$ in the latter polyhedra.

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