Kourovka Notebook Problem 21.139

Definitions and mathematical statement

A finitely generated group $G$ is word-hyperbolic if its Cayley graph for a finite generating set has some $\delta\geq0$ such that every geodesic triangle has each side in the $\delta$-neighborhood of the other two sides. A right-angled Artin group has generators the vertices of a finite graph $\Lambda$, with commuting relations exactly on its edges. Its Salvetti cube complex has one vertex, one loop per generator, and a coordinate torus for each complete subgraph, glued along the corresponding coordinate subtori.

Use the standard equivalent description of a compact special cube complex: a compact nonpositively curved cube complex admitting a cubical local isometry to a Salvetti complex. A cube complex glues unit Euclidean cubes along faces; nonpositive curvature means every vertex link is flag (every finite clique spans a simplex). A cubical local isometry induces an injective map of links onto a full subcomplex, containing every simplex spanned by its vertices. The group $G$ is virtually compact special if a finite-index subgroup is the fundamental group of such a complex. For $H\leq G$ let $b_2(H)=\dim_{\mathbb Q}H_2(H;\mathbb Q)$ be its second rational group-homology Betti number, computed as the homology of a classifying space $K(H,1)$ (a connected CW-complex with fundamental group $H$ and contractible universal cover).

Suppose $G$ is hyperbolic and virtually compact special, and that \[ \exists B<\infty\quad\forall H\leq G\text{ of finite index},\quad b_2(H)\leq B. \] Must $G$ have a finite-index subgroup that is a free group or the fundamental group of a closed surface of negative Euler characteristic? Surfaces with boundary have free fundamental groups and are included in the first alternative. The bound must hold for all finite-index subgroups, not just one tower of covers.

Short English statement

Does uniformly bounded second homology in finite covers force a hyperbolic virtually compact special group to be virtually free or a surface group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2648 (KOU-21.139), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.139. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.