Kirby Problem 3.22

Definitions and mathematical statement

Let $M$ be a connected complete finite-volume hyperbolic three-manifold and $G=\pi_1(M)$. A faithful action on a one-manifold $Y$ is an injective homomorphism $G\to\operatorname{Homeo}(Y)$. For an integer $r\geq1$, a $C^r$ action has image in $\operatorname{Diff}^r(Y)$: each map and its inverse has $r$ continuous derivatives. A $C^{k,\alpha}$ action additionally requires the $k$th derivatives to be Hölder continuous with exponent $0<\alpha\leq1$. Smooth means $C^\infty$.

Determine the greatest regularity attainable by faithful actions of $G$ on $S^1$, and on the compact interval $[0,1]$. For the latter, differentiability is required up to the endpoints, with one-sided derivatives there. Also ask the virtual version: replace $G$ by some finite-index subgroup $H\leq G$, which may be chosen to obtain the highest possible regularity.

The question concerns existence of an action, rather than regularity of one fixed construction. A related refinement asks whether natural topological actions can be conjugated to smoother ones. Compactness of the interval and endpoint regularity matter: a smooth action on the open line need not extend smoothly to a compact interval. If no action attains a proposed supremal regularity, its nonattainment should be distinguished from existence at that level.

Short English statement

How smooth can a faithful circle or compact-interval action of a hyperbolic three-manifold group be, possibly after passing to a finite-index subgroup?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2820 (KP-3.22), 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 3.22. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.