Kirby Problem 3.29

Definitions and mathematical statement

Let $M$ be a closed connected hyperbolic three-manifold, given by a finite triangulation with $t$ tetrahedra. A connected finite-sheeted covering $p:N\to M$ has degree $d=[\pi_1(M):p_*\pi_1(N)]$. Its first Betti number is $b_1(N)=\dim_{\mathbb Q}H_1(N;\mathbb Q)$. Define \[ d_b(M)=\min\{\deg(p):b_1(N)>0\},\qquad d_f(M)=\min\{\deg(p):N\text{ fibers over }S^1\}. \] Fibering means that $N$ is a mapping torus of a homeomorphism of a closed surface. These minima are finite by virtual fibering, and $d_b(M)\leq d_f(M)$.

Find explicit upper bounds for these quantities from finite data for $M$. In particular, do constants $C,k>0$, independent of $M$ and its triangulation, exist such that $d_b(M)\leq Ct^k$ whenever $M$ is triangulated with $t$ tetrahedra? Ask separately for a useful explicit bound on $d_f(M)$; the source does not automatically demand that this second bound be polynomial.

The existence of some suitable cover is already part of the setup. An unrestricted enumeration of finite covers that eventually finds one does not supply the requested a priori quantitative degree estimate.

Short English statement

Give explicit bounds on the degrees of covers of closed hyperbolic three-manifolds with positive first Betti number or a surface fibration, and test a polynomial bound for the former.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2827 (KP-3.29), 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 3.29. Expanded formulation of the supplied catalogue statement. https://math.berkeley.edu/sites/default/files/surv-295-ruberman-watermarked-author-pdf.pdf.