Kirby Problem 2.17

Definitions and mathematical statement

Let $S$ be a closed oriented surface of genus at least two. A geodesic current on $S$ is a $\pi_1(S)$-invariant Radon measure on the space of unoriented geodesics in its hyperbolic universal cover, equivalently on unordered pairs of distinct boundary points. A nontrivial closed curve $\alpha$, taken up to free homotopy, defines a counting current $[\alpha]$ by counting its lifts; iterates count with their multiplicities.

For each marked hyperbolic metric $X$ on $S$, a current $c$ has a length $\ell_X(c)=i(c,L_X)$. Here $L_X$ is the Liouville current and $i$ is the continuous extension of geometric intersection number, normalized so that $i([\alpha],L_X)$ is the length of the geodesic representative of $\alpha$.

Suppose there is a closed curve $\gamma$ for which \[ \ell_X(c)=\ell_X([\gamma])\quad\hbox{for every marked hyperbolic metric }X. \] Must $c$ be a convex combination of closed-curve currents? Explicitly, must there be finitely many closed curves $\alpha_1,\ldots,\alpha_r$ and real numbers $t_j\geq0$ with $\sum_jt_j=1$ and $c=\sum_jt_j[\alpha_j]$? The conclusion concerns equality as measures, not only equality of lengths for one metric. The curves need not be simple.

Short English statement

If a current has exactly the same length as one closed curve in every hyperbolic metric, must it be a finite convex mixture of closed curves?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2765 (KP-2.17), 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 2.17. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.