Questions in Geometric Group Theory — Q 12.3

Definitions and mathematical statement

Let $\Sigma_g$ denote a closed connected orientable surface of genus $g\geq2$. Its mapping class group $\operatorname{Mod}(\Sigma_g)$ consists of orientation-preserving homeomorphisms modulo isotopy. A pseudo-Anosov class has a representative preserving a pair of transverse singular measured foliations, stretching one transverse measure by $\lambda>1$ and contracting the other by $\lambda^{-1}$.

The source's part (a) asks for an injective homomorphism from some $\pi_1(\Sigma_g)$ to some $\operatorname{Mod}(\Sigma_h)$, $g,h\geq2$, whose nonidentity image elements are all pseudo-Anosov. This existence assertion is answered affirmatively by Kent--Leininger [S3], and is recorded as context rather than as an unresolved target.

Part (b) asks whether there exist $g,h\geq2$ and a smooth surface bundle \[ \Sigma_h\longrightarrow M\longrightarrow\Sigma_g \] whose fundamental group is word-hyperbolic. A bundle is locally a product with transition maps diffeomorphisms of the fiber; its total space is a closed four-manifold. Word-hyperbolicity means that some Cayley graph of the fundamental group has uniformly thin geodesic triangles.

The stronger geometric version asks whether such an $M$ can admit a Riemannian metric of constant sectional curvature $-1$. Both genera and the bundle may vary. The existence of the purely pseudo-Anosov representation in (a) does not itself assert either conclusion in (b). In particular, absence of a subgroup $\mathbb Z^2$ alone is not the definition of hyperbolicity. A metric satisfying the stronger target would imply the group-theoretic target.

Short English statement

Can a closed surface bundle over a closed surface have hyperbolic fundamental group, or even a real hyperbolic metric?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101203 (AMR-010-1203), 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 12.3 (PDF page 19). Original source indexed by AMR-010-1203. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.
• S3 Autumn E. Kent and Christopher J. Leininger, Atoroidal surface bundles. Purely pseudo-Anosov surface subgroups. https://annals.math.princeton.edu/articles/22768.