Kirby Problem 2.11

Definitions and mathematical statement

Let $S$ be a connected orientable finite-type hyperbolic surface and $\gamma$ an essential nonperipheral closed curve, considered up to free homotopy. Its simple lifting degree $d_S(\gamma)$ is the least degree of a connected finite-sheeted cover $p:S'\to S$ in which the map $\gamma:S^1\to S$ lifts to a simple closed curve. The lift must project to the specified curve itself; replacing it by a power is not allowed. Let $i(\gamma,\gamma)$ be its minimal self-intersection number, and let $\ell_\rho(\gamma)$ be its geodesic length for a fixed complete hyperbolic metric $\rho$.

Find sharp bounds for the extremal functions \[ f_S(n)=\sup_{i(\gamma,\gamma)\leq n}d_S(\gamma),\qquad f_\rho(L)=\sup_{\ell_\rho(\gamma)\leq L}d_S(\gamma). \] Seek analogous bounds using conjugacy word length in a fixed finite generating set of $\pi_1(S)$. Determine also typical or average growth, specifying the sampling model, for example uniform sampling from the finite set of closed geodesic classes of length at most $L$. The surface, metric, and generating set must remain fixed when taking the relevant asymptotic limit. Sharp average bounds need not coincide with extremal ones.

Short English statement

How large a finite cover is needed to turn a closed surface curve into a simple curve, in the worst case and on average?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2759 (KP-2.11), 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 2.11. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.