5.4 (Agol) — Virtual CAT(0) cubical manifold models

Definitions and mathematical statement

Question (Agol). Does every hyperbolic $3$-manifold $M$ have a finite-sheeted covering space $\widetilde M$ homeomorphic to a CAT(0) cube complex? In literal quantified form, ask whether there exist a finite-sheeted covering $p:\widetilde M\to M$, a cubical complex $X$ with its intended CAT(0) metric, and a homeomorphism $\widetilde M\cong X$, for each $M$ in the intended class. The source does not fix compactness, completeness, finite volume, boundary, or orientation conventions. It suggests arithmetic $3$-manifolds containing a geodesic surface as a test case.

There is a necessary terminology qualification. In standard metric terminology a globally CAT(0) space is contractible. A locally CAT(0), or nonpositively curved, cube complex can have nontrivial fundamental group; its universal cover is CAT(0) under the usual hypotheses. These are different conditions. For example, a finite cover of a closed hyperbolic $3$-manifold has nontrivial fundamental group and so cannot be homeomorphic to a globally CAT(0) space. Thus the literal global-CAT(0) wording conflicts with the intended inclusion of such manifolds. A possibly intended question asks for a locally CAT(0) cubical structure on the finite cover, but this is a distinct formulation and is not silently substituted.

The supplied summary describes virtual-specialness-type results, credits Agol and Haglund--Wise, and mentions finite-index fundamental groups inside right-angled Coxeter groups acting on CAT(0) cube complexes. Such group actions or cubical classifying spaces do not by themselves exhibit a cube complex homeomorphic to the covering manifold. The summary calls the stronger wording open and says the relevant cover is a special cube complex; its scope and use of "cover" versus group model require review in light of the global/local distinction.

Short English statement

The source asks for a finite cover of every hyperbolic $3$-manifold that is homeomorphic to a CAT(0) cube complex. Its wording needs clarification: global CAT(0) implies contractibility, whereas the plausible cubical-manifold requirement may be local CAT(0). Group cubulations alone do not settle the homeomorphism requirement.

Sources

• Local UnsolvedMath registry, permanent ID 10900029, source code catalogue entry; attribution Agol, subsection 5.4, "Virtual CAT(0) cubical manifold models."
• Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), subsection 5.4, PDF page 4. Supplied URLs: https://arxiv.org/pdf/1512.04620#page=4 and https://arxiv.org/abs/1512.04620; recorded access 2026-07-29; extraction source-tex checked against rendered PDF text. The source proposes arithmetic $3$-manifolds containing a geodesic surface as a test case.
• The supplied summary credits Agol's virtual specialness theorem and Haglund--Wise, without full bibliographic entries, and reports no verified solution to its stronger homeomorphism formulation. Its statements are retained as supplied assessments requiring the metric and model-space distinctions above.
• Supplied status open and OPEN-TRIAGE. The source presents the item as open in 2015; no complete modern primary-literature audit is recorded. Status evidence and redistribution terms of the publicly accessible author-submitted TeX and PDF remain NEEDS_REVIEW.