Lackenby's 2006 program relating Heegaard splittings, spectral geometry, and virtual structure of hyperbolic 3-manifolds produced two conjectures about how Heegaard complexity can degenerate in finite covers: (1) a compact orientable hyperbolic 3-manifold has zero Heegaard gradient if and only if it virtually fibers over the circle, and (2) every closed orientable hyperbolic 3-manifold has strictly positive strong Heegaard gradient. Conjecture (1) is now a theorem, an immediate corollary of Agol's virtual fibering theorem together with the elementary observation that fibered covers give bounded Heegaard genus. The surviving open core is Conjecture (2): the minimal complexity of a strongly irreducible Heegaard splitting must grow at least linearly with covering degree in every finite cover of every closed orientable hyperbolic 3-manifold. The best unconditional bounds are Renard's double-logarithmic lower bound for arbitrary covers and Lackenby's square-root bound for regular covers; neither a proof nor a refutation is known.

Background

A Heegaard splitting of a closed orientable 3-manifold MM is a decomposition M=V∪SWM = V \cup_S W in which SS is a closed orientable surface and VV, WW are handlebodies (balls with 1-handles attached); the minimal genus g(M)g(M) of such a surface SS is the Heegaard genus of MM. It is convenient to work with the Heegaard Euler characteristic χ−h(M)=2g(M)−2\chi^h_-(M) = 2g(M) - 2 for connected MM. A splitting is strongly irreducible (Casson–Gordon) if every properly embedded meridian disk of VV intersects every meridian disk of WW; this rules out the splitting being decomposable into simpler layered pieces. The strong Heegaard Euler characteristic χ−sh(M)\chi^{sh}_-(M) is the negative Euler characteristic of a minimal-genus strongly irreducible Heegaard surface, with the convention χ−sh(M)=+∞\chi^{sh}_-(M) = +\infty if MM has no strongly irreducible splitting; thus χ−sh(M)≥χ−h(M)\chi^{sh}_-(M) \geq \chi^h_-(M). A closed hyperbolic 3-manifold is a closed 3-manifold whose interior carries a complete Riemannian metric of constant sectional curvature −1-1; by Mostow rigidity (the theorem that a finite-volume hyperbolic manifold of dimension at least 3 is determined up to isometry by its fundamental group) its topology alone determines this geometry. Given a collection of finite covers Mi→MM_i \to M of degrees did_i, Lackenby defined the infimal Heegaard gradient as inf⁡iχ−h(Mi)/di\inf_i \chi^h_-(M_i)/d_i and the infimal strong Heegaard gradient as lim inf⁡iχ−sh(Mi)/di\liminf_i \chi^{sh}_-(M_i)/d_i; when no collection is specified, all finite covers are understood. These invariants measure how fast Heegaard complexity must grow, on average, as one passes to covers — a topological analogue of the rank gradient and cost of a group (group-theoretic invariants measuring, over finite-index subgroups, the asymptotic growth of the minimal number of generators and of the minimal number of edges in a graph carrying the fundamental group, respectively). A basic example drives the subject: if MM fibers over the circle with fiber a surface FF, then the covers obtained by unwrapping the base circle still fiber with the same fiber and satisfy χ−h≤2∣χ(F)∣+4\chi^h_- \leq 2|\chi(F)| + 4, so any virtually fibered manifold (one having a finite cover that fibers over the circle) has zero Heegaard gradient. In this language Lackenby proposed, in his study of the (then-open) virtually Haken conjecture (that every closed hyperbolic 3-manifold has a finite cover containing a properly embedded, π1\pi_1-injective surface) and the virtually fibered conjecture (that every closed hyperbolic 3-manifold has a finite cover that fibers over the circle), via Property (τ\tau) — the requirement that a family of finite-index subgroups of a group have a uniform spectral gap for the averaging operators on the coset graphs, which forces the Cheeger constants (isoperimetric constants) of the corresponding covers to be bounded below: the Heegaard gradient conjecture — a compact orientable hyperbolic 3-manifold has zero Heegaard gradient if and only if it virtually fibers over the circle (a revision prompted by Ichihara's computation of Heegaard gradients of Seifert fibered manifolds) — and the strong Heegaard gradient conjecture — every closed orientable hyperbolic 3-manifold has positive strong Heegaard gradient. The landscape changed with Agol's theorem that every closed hyperbolic 3-manifold is virtually fibered (via the theory of virtually special cube complexes — cube complexes, that is, spaces glued from Euclidean cubes, admitting a finite cover whose combinatorial hyperplanes behave in a highly controlled way — building on Wise, Haglund, Kahn–Markovic and others), so the first conjecture became a corollary: virtual fibering supplies bounded-genus fibered covers, making the Heegaard gradient of every finite-volume hyperbolic 3-manifold equal to zero. The strong conjecture, by contrast, was sharpened rather than settled by this development: it now asserts that even though every closed hyperbolic 3-manifold admits towers of covers whose ordinary Heegaard genus is bounded, any strongly irreducible Heegaard splitting of any sufficiently large finite cover must have complexity growing at least linearly in the covering degree. Known supporting results are quantitative but far from linear: for finite regular covers of a closed negatively curved 3-manifold the strong Heegaard genus cannot grow more slowly than the square root of the degree (Lackenby), Property (τ\tau) methods give positive gradient only along selected cofinal regular towers (Long–Lubotzky–Reid; Jeon for congruence towers, that is, towers of regular covers obtained by reducing an arithmetic description of the fundamental group modulo increasing integers), and Renard's minimal-surface methods (using surfaces of least area for the ambient hyperbolic metric) yield, for arbitrary covers of a closed hyperbolic manifold, an explicit constant kˉ>0\bar{k} > 0 with kˉ χ−sh(M′)ln⁡χ−sh(M′)>ln⁡ln⁡d\bar{k}\,\chi^{sh}_-(M')\ln \chi^{sh}_-(M') > \ln\ln d — a double-logarithmic lower bound, equivalent to positivity of 'sub-logarithmic' gradient variants she introduced.

Problem Statement

Prove or refute Lackenby's strong Heegaard gradient conjecture: for every closed orientable hyperbolic 3-manifold MM, the strong Heegaard gradient

∇sh(M)  =  lim inf⁡[M′:M]→∞χ−sh(M′)[M′:M]\nabla^{sh}(M) \;=\; \liminf_{[M':M] \to \infty} \frac{\chi^{sh}_-(M')}{[M':M]}

is strictly positive, where the liminf is taken over all finite covers M′→MM' \to M of degree [M′:M][M':M] and χ−sh(M′)\chi^{sh}_-(M') denotes the negative Euler characteristic of a minimal-genus strongly irreducible Heegaard splitting of M′M' (equal to +∞+\infty if M′M' admits no strongly irreducible Heegaard splitting). Equivalently: there exist constants c>0c > 0 and D<∞D < \infty, depending on MM, such that every finite cover M′→MM' \to M of degree at least DD with a strongly irreducible Heegaard splitting satisfies χ−sh(M′)≥c [M′:M]\chi^{sh}_-(M') \geq c\,[M':M]. A refutation consists of an explicit closed orientable hyperbolic MM and a sequence of finite covers Mi→MM_i \to M of degrees di→∞d_i \to \infty admitting strongly irreducible Heegaard splittings with χ−sh(Mi)/di→0\chi^{sh}_-(M_i)/d_i \to 0. Note the conjecture does not assert that all covers admit strongly irreducible splittings — a cover with none contributes +∞+\infty and does not lower the gradient; the claim is precisely that strongly irreducible Heegaard complexity grows at least linearly with covering degree across the full family of finite covers. (The companion Heegaard gradient conjecture, characterizing virtual fibering by vanishing of the ordinary gradient, is settled by Agol's virtual fibering theorem and is not part of this problem; Lackenby stated the strong conjecture for closed orientable hyperbolic manifolds, which is the scope fixed here.)

References

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