Convex cocompact right-angled Gromov-Thurston polyhedra

Abstract: Lee and Marquis exhibited convex cocompact hyperbolic reflection groups in dimension 5 whose limit sets are homeomorphic to the 3-sphere, but none of whose finite-index subgroups can be realized as 4-dimensional real hyperbolic lattices. Using different methods, we furnish right-angled examples.

• The paper constructs convex cocompact right-angled reflection groups in Isom(5) whose limit sets are homeomorphic to S3, under a new and uniform approach.
• Employing Mostow-Regularity the paper shows that no finite-index subgroups of these reflection groups can be discretely embedded in the lower dimensional Isom(4)
• The analysis covers the absence of such groups in dimensions lower than 5 and applies a rigorous process for constructing convex cocompact right-angled reflection subgroups in arithmetic contexts with 3-dimensional Sierpinski limit sets

Overview and main result

This note by Sami Douba constructs convex cocompact right-angled reflection groups in $\mathrm{Isom}(\mathbb{H}^5)$ whose limit sets are homeomorphic to $S^3$, yet none of whose finite-index subgroups embed discretely in $\mathrm{Isom}(\mathbb{H}^4)$ (2608.19640). The result refines an earlier construction of Lee–Marquis, who produced such groups without the right-angled property, using deformations of Esselmann's compact 4-dimensional hyperbolic polyhedra; their argument depends essentially on mutually intersecting walls. The present work instead follows a strategy suggested by Gromov–Thurston: starting from a compact right-angled polyhedron $P \subset \mathbb{H}^4$ (all known examples are commensurable to the right-angled 120-cell), one replaces two adjacent orthogonal walls $H_1, H_2$ with round hyperspheres meeting at angle $\pi/2n$, obtaining a Coxeter group $W^n(P,H_1,H_2)$ realized as a convex cocompact reflection group in $\mathrm{PO}(5,1)$. Geometrically, $R_n$ is an $n$-fold cyclic cover of the original reflection orbifold branched over the codimension-2 face $S^3$0—hence the "Gromov–Thurston" designation.

The main theorem has a sharp dimensional scope. By Andreev's theorem together with Bestvina–Mess and Davis, no analogous examples exist one dimension lower; conversely, by Bestvina–Mess and Januszkiewicz–Świątkowski there is no abstract Gromov-hyperbolic right-angled Coxeter group whose Gromov boundary is a sphere of dimension $S^3$1, so the theorem cannot be pushed upward either.

Rigidity obstruction

The non-embedding half of the theorem rests on two lemmas. First, if $S^3$2 is a finite-index subgroup of $S^3$3 with trivial centralizer in $S^3$4 and $S^3$5 embeds as a lattice in $S^3$6 for $S^3$7, then so does $S^3$8: Mostow–Prasad rigidity supplies, for each $S^3$9, a unique isometry conjugating $\mathrm{Isom}(\mathbb{H}^4)$0 to its $\mathrm{Isom}(\mathbb{H}^4)$1-conjugate, yielding an extension of $\mathrm{Isom}(\mathbb{H}^4)$2 whose kernel is precisely the centralizer. Second, every finite-index subgroup of an irreducible, non-virtually-abelian Coxeter group has trivial centralizer, via the faithful Zariski-dense Tits–Vinberg representation into some $\mathrm{Isom}(\mathbb{H}^4)$3 furnished by Benoist–de la Harpe and de Cornulier.

Combining these, Proposition (nothyperbolic) shows that for any $\mathrm{Isom}(\mathbb{H}^4)$4, no finite-index subgroup of $\mathrm{Isom}(\mathbb{H}^4)$5 embeds discretely in $\mathrm{Isom}(\mathbb{H}^4)$6. The proof is a rigidity-theoretic "interbreeding" argument: assuming a discrete embedding, Mostow rigidity forces $\mathrm{Isom}(\mathbb{H}^4)$7 itself to be a cocompact lattice, hence the Coxeter data are realized by a compact polyhedron $\mathrm{Isom}(\mathbb{H}^4)$8 with dihedral angles $\mathrm{Isom}(\mathbb{H}^4)$9. One then builds compact right-angled polyhedra $P \subset \mathbb{H}^4$0 and $P \subset \mathbb{H}^4$1 from $P \subset \mathbb{H}^4$2 and $P \subset \mathbb{H}^4$3, glues them along a common compact right-angled 3-polyhedron facet using Mostow rigidity in dimension 3, and obtains a compact right-angled polyhedron $P \subset \mathbb{H}^4$4 with an abstract automorphism $P \subset \mathbb{H}^4$5 of its reflection group that is inner by $P \subset \mathbb{H}^4$6 on one Zariski-dense subgroup and inner by $P \subset \mathbb{H}^4$7 on another. Mostow rigidity in dimension 4 identifies both products, forcing $P \subset \mathbb{H}^4$8; but the former has order $P \subset \mathbb{H}^4$9 while the latter has order 2, so $H_1, H_2$0. A notable strengthening over the classical Gromov–Thurston argument is uniformity: the threshold beyond which the conclusion holds is $H_1, H_2$1 uniformly in $H_1, H_2$2, whereas Gromov–Thurston's argument gives only some $H_1, H_2$3.

Convex cocompact realization

The existence half constructs, for each $H_1, H_2$4, a compact right-angled polyhedron $H_1, H_2$5 and orthogonal walls $H_1, H_2$6 such that $H_1, H_2$7 embeds as a convex cocompact reflection group in $H_1, H_2$8. Starting from the right-angled 120-cell tiling of $H_1, H_2$9, one takes metric $\pi/2n$0-neighborhoods $\pi/2n$1 of the walls chosen so that $\pi/2n$2 meets $\pi/2n$3 at angle $\pi/2n$4 (existence by the intermediate value theorem), builds a compact right-angled polyhedron $\pi/2n$5 from coarse convex hulls within the tiling, and then—in the Poincaré ball model viewed inside $\pi/2n$6—replaces $\pi/2n$7 by the round hyperspheres $\pi/2n$8. The Poincaré polyhedron theorem identifies the resulting inversion group with $\pi/2n$9, and convex cocompactness follows because the doubled polyhedron $W^n(P,H_1,H_2)$0 is right-angled with no asymptotic walls, so its reflection group is convex cocompact; $W^n(P,H_1,H_2)$1 is a finite-index supergroup thereof.

Since the nerve of $W^n(P,H_1,H_2)$2 coincides with that of $W^n(P,H_1,H_2)$3—a triangulation of $W^n(P,H_1,H_2)$4 dual to the surface of $W^n(P,H_1,H_2)$5—the Davis complex remains homeomorphic to $W^n(P,H_1,H_2)$6 with CAT(0) boundary $W^n(P,H_1,H_2)$7, and virtual cohomological dimension stays 4. Convex cocompactness then transfers this boundary identification equivariantly to the limit set in $W^n(P,H_1,H_2)$8, completing the theorem. By known arguments (Lee–Marquis §6), any such group fails to virtually embed as a lattice in any semisimple real algebraic group and is not quasiisometric to any symmetric space.

Arithmetic subgroups and Sierpiński limit sets

A variant of the construction, taking the walls $W^n(P,H_1,H_2)$9 at angle $\mathrm{PO}(5,1)$0 rather than $\mathrm{PO}(5,1)$1, places the resulting groups inside a cocompact arithmetic subgroup of $\mathrm{PO}(5,1)$2—a conjugate of $\mathrm{PO}(5,1)$3 for the golden ratio $\mathrm{PO}(5,1)$4 and the quadratic form $\mathrm{PO}(5,1)$5. Varying the neighborhoods $\mathrm{PO}(5,1)$6 with $\mathrm{PO}(5,1)$7 fixed yields an abundance of Gromov–Thurston reflection subgroups of this fixed lattice, probably infinitely many up to wide commensurability. Although the non-embedding proposition assumed even $\mathrm{PO}(5,1)$8, the symmetry of the 120-cell ensures it still applies here despite the odd submultiple angle.

Arithmeticity enables an application via Bergeron–Haglund–Wise separability and Scott's criterion: cutting a finite cover along a boundary component of the convex core and reattaching ends produces convex cocompact subgroups $\mathrm{PO}(5,1)$9 whose limit sets are 3-dimensional Sierpiński compacta, but whose peripheral sphere stabilizers admit no discrete embedding (even virtually) into $R_n$0 stabilizing a round 3-sphere. This contrasts sharply with dimension 3, where McMullen showed any convex cocompact subgroup of $R_n$1 with Sierpiński curve limit set admits a convex cocompact representation with all peripheral circles round. Independently, the author observes that the $R_n$2 diagram ($R_n$3) from Lee–Marquis also lies in a cocompact arithmetic subgroup—the Galois conjugates of its Gram matrix under both nontrivial embeddings of $R_n$4 being positive-definite—and provides an alternative source for such $R_n$5.

Limitations and open questions

Several points remain open or conditional. The paper does not determine whether the Lee–Marquis examples admit finite-index right-angled reflection subgroups, nor whether an argument like the interbreeding proof rules out discrete embeddings of those examples into $R_n$6—the obstacle being the abundance of angles that are odd submultiples of $R_n$7 in Esselmann's polyhedra. The suggested route through Hamenstädt–Jäckel's results on negatively curved metrics on Gromov–Thurston manifolds is indicated as likely but not carried out. In higher dimensions, the author expects the role of compact right-angled hyperbolic polyhedra (which cease to exist above dimension 4) can be replaced by Bergeron–Haglund–Wise separability to produce closed aspherical manifolds of each dimension $R_n$8 admitting flat conformal structures with convex cocompact holonomy but not homotopy equivalent to any compact locally symmetric space—a claim announced by M. Kapovich, though plausibly predating the separability machinery—but this is not proven here. Finally, the Hausdorff dimension of the limit set $R_n$9 can presumably be made arbitrarily close to 3 by enlarging the neighborhoods $n$0, though Yue's theorem and the non-embedding result force it to remain strictly above 3; the exact infimum is not computed.

Conclusion

The paper supplies right-angled examples of convex cocompact reflection groups in $n$1 with $n$2 limit sets that are virtually indiscrete in $n$3, achieved through a uniform-in-$n$4 rigidity argument combined with an explicit Poincaré polyhedron theorem construction branching over codimension-2 faces of the 120-cell. The arithmetic refinement additionally yields convex cocompact subgroups with Sierpiński compactum limit sets whose peripheral stabilizers resist round-sphere embeddings, exhibiting a qualitative failure of a dimension-3 flexibility phenomenon in dimension 5.