Questions in Geometric Group Theory — Q 1.18

Definitions and mathematical statement

Let $G$ be given by a finite presentation, with the promise that it is word-hyperbolic: its Cayley graph has uniformly thin geodesic triangles. Its Gromov boundary $\partial G$ is the compact space of geodesic rays from a basepoint, with rays identified when they remain a bounded distance apart. Write $\check H^i(\partial G;\mathbb Z)$ for integral Čech cohomology, computed as the direct limit of the cohomology of nerves of finite open covers under refinement.

Find an algorithm that determines these cohomology groups from the presentation. A concrete decision problem, avoiding an unspecified output format for arbitrary abelian groups, is whether there is an algorithm which always halts and decides \[ \check H^i(\partial G;\mathbb Z)\cong \check H^i(S^2;\mathbb Z)\quad\hbox{for every }i\geq0. \] Thus the unreduced groups sought are $\mathbb Z$ in degrees zero and two, and zero in all other degrees. In reduced notation the only nonzero group is $\widetilde{\check H}^{2}\cong\mathbb Z$.

The comparison with group-ring cohomology in the source uses reduced boundary cohomology; retaining that convention avoids a spurious degree-zero term. The input is finite algebraic data, not an already supplied triangulation of the boundary. The boundary can be a nonmanifold compactum, so computing ordinary simplicial cohomology of a single finite complex is not assumed to suffice.

Short English statement

Can one algorithmically compute boundary cohomology, and in particular recognize the integral cohomology of the two-sphere?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100118 (AMR-010-0118), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 1.18 (PDF page 4). Original source indexed by AMR-010-0118. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.