Kirby Problem 2.12

Definitions and mathematical statement

Let $S_g$ be a closed connected orientable surface of genus $g\geq2$. For a hyperbolic metric $\rho$ on $S_g$, a systole is an unoriented closed geodesic of least positive length. Count distinct geometric geodesics, without choices of starting point or orientation. Define \[ N(g)=\max_{\rho}\#\{\text{systoles of }(S_g,\rho)\}. \] Determine $N(g)$, in particular its optimal asymptotic order as $g$ grows.

The associated topological question asks for \[ C(g)=\max\{|\mathcal A|: \mathcal A\text{ consists of distinct essential simple curve classes on }S_g,\ i(a,b)\leq1\}. \] Here essential means not null-homotopic; $i(a,b)$ is the smallest number of transverse intersections among representatives of the two isotopy classes. Require the inequality for every two different members of $\mathcal A$. Determine $C(g)$ exactly or asymptotically, with matching upper bounds and constructions. The curve-set question also makes sense in genus one. Systoles are simple and any two intersect at most once on a closed hyperbolic surface, so the first problem is a geometrically restricted case of the second. No metric is prescribed in maximizing $N(g)$.

Short English statement

What are the largest possible collections of systoles, and of simple curves meeting pairwise at most once, on a genus-$g$ surface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2760 (KP-2.12), Kirby's Problems in Low-Dimensional Topology. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 K3: A New Problem List in Low-Dimensional Topology, Problem 2.12. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.