Positive first Betti number of finite covers of random Heegaard splittings (Dunfield-Thurston Questions 4.8 and 9.2 and the non-solvable extension of Theorem 9.1)

Dunfield and Thurston's 2006 model of a random 3-manifold glues two genus-g handlebodies by the endpoint of a length-L random walk on the mapping class group of the genus-g surface. The paper computed how often the resulting manifold has finite covers of prescribed kinds, and its abstract singles out as 'the next natural question' whether those covers have positive first Betti number beta_1. Later work sharpened both sides of this question: Maher proved the random splitting is hyperbolic with asymptotic probability 1 and Viaggi proved its hyperbolic volume grows linearly in L, so by Agol's virtual fibered theorem the random splitting has some finite cover with beta_1 > 0 with asymptotic probability 1, but only of unbounded degree; on the other side, the probability that a cover with a fixed solvable covering group Q (Dunfield-Thurston for genus 2 and abelian Q, Rivin for all solvable Q), or any cover of bounded degree in the large-genus regime (Sawin-Wood), has beta_1 > 0 is known to be 0. (i) for a fixed non-solvable finite group Q at fixed genus, is the probability of a Q-cover with beta_1 > 0 also 0, or what is its value; (ii) Dunfield-Thurston Question 9.2, what is the probability that the maximal abelian cover of the random splitting, whose deck group H_1(M;Z) is typically finite but of expected order growing with L, has beta_1 > 0; (iii) Dunfield-Thurston Question 4.8, whether the profinitefold group of a Haar-random profinite Heegaard diagram almost surely has a finite-index subgroup surjecting onto the profinite integers (Sawin-Wood resolved the reordered large-genus analogue negatively).

Background

Every closed orientable 3-manifold $M$ admits a Heegaard splitting: it is obtained by gluing two copies of a genus- $g$ handlebody (a regular neighborhood of a wedge of $g$ circles in 3-space) along their common boundary, the closed orientable surface $\Sigma_g$ of genus $g$ , by a homeomorphism $\varphi$ . The isotopy class of $\varphi$ is an element of the mapping class group $\mathrm{Mod}(\Sigma_g)$ , the group of isotopy classes of self-homeomorphisms of $\Sigma_g$ . Dunfield and Thurston introduced a probabilistic model of 3-manifolds by taking $\varphi$ to be the endpoint of a random walk of length $L$ on $\mathrm{Mod}(\Sigma_g)$ with respect to a fixed finite symmetric generating set; denote the resulting manifold by $M_{g,L}$ . They proved that for the properties relevant here the limiting probabilities as $L \to \infty$ exist and are independent of the chosen generating set, and they also studied a large-genus regime in which $L \to \infty$ and then $g \to \infty$ .

The original motivation was the Virtual Haken Conjecture, which posited that every irreducible 3-manifold (one in which every embedded 2-sphere bounds a 3-ball) with infinite fundamental group has a finite cover that is Haken, i.e. contains a properly embedded surface on which $\pi_1$ injects. A finite cover with positive first Betti number $\beta_1 = \mathrm{rank}\, H_1(\cdot;\mathbb{Z})$ is Haken, and the conjecture, indeed the stronger statement that every hyperbolic 3-manifold is virtually fibered (finitely covered by a bundle over the circle), was proved by Agol in 2013.

Dunfield and Thurston computed how often a random splitting has a finite cover of a prescribed type. A $Q$ -cover, for a finite group $Q$ , is a regular covering whose group of deck (covering) transformations is $Q$ ; equivalently, an epimorphism $\pi_1(M) \to Q$ considered up to automorphisms of $Q$ . For $Q$ a non-abelian finite simple group they proved that as $g \to \infty$ the probability of a $Q$ -cover converges to $1 - e^{-\mu}$ , where $\mu = |H_2(Q;\mathbb{Z})|/|\mathrm{Out}(Q)|$ with $H_2(Q;\mathbb{Z})$ the second integral homology group of $Q$ (its Schur multiplier) and $\mathrm{Out}(Q)$ its outer automorphism group, and the number of $Q$ -covers converges to a Poisson random variable of mean $\mu$ . Random 3-manifolds thus have far more finite quotients than groups defined by random balanced presentations (presentations with as many relators as generators). Maher later proved the hyperbolicity half of their Conjecture 2.11, Viaggi proved the volume half in the strong form $\mathrm{vol}(M_{g,L})/L \to v > 0$ almost surely, and Sawin and Wood determined the full limiting distribution of the profinite completion $\widehat{\pi_1(M_{g,L})}$ (the inverse limit of all finite quotients) in the large-genus regime, settling the general- $Q$ distribution question that Dunfield and Thurston left open.

The question that survives is exactly the one the source abstract calls the next natural question: do these covers have positive $\beta_1$ ? The tension is sharp. By Maher's hyperbolicity theorem combined with Agol's virtual fibered theorem, a random splitting virtually fibers with asymptotic probability 1, so it has some finite cover with $\beta_1 > 0$ , but only of degree tending to infinity with $L$ . On the other side, Dunfield and Thurston proved (Theorem 9.1) that for a fixed abelian group $Q$ and genus 2 the probability that a $Q$ -cover has $\beta_1 > 0$ is 0; Rivin extended this to every solvable $Q$ at every fixed genus $g \ge 2$ , with exponential decay in $L$ ; and Sawin and Wood proved that in the large-genus regime the probability of any cover of degree at most $n$ with $\beta_1 > 0$ tends to 0, and that their limiting random profinite group almost surely has no open subgroup surjecting onto $\widehat{\mathbb{Z}}$ .

What remains unknown is the regime where the covering group is allowed to grow with the manifold. The maximal abelian cover of $M_{g,L}$ is the cover with deck group $H_1(M_{g,L};\mathbb{Z})$ , the abelianization of $\pi_1$ ; for a random splitting this group is typically finite, but Dunfield and Thurston showed its expected order grows with $L$ , so the maximal abelian cover is not a fixed- $Q$ or fixed-degree family: asking whether it has $\beta_1 > 0$ asks whether $H_1(M;\mathbb{Q}_\chi) \ne 0$ for some character $\chi$ of $H_1(M;\mathbb{Z})$ , possibly of very large order. The profinite analogue, Question 4.8 of the paper, concerns the profinitefold group: the quotient of the profinite completion of $\pi_1(\Sigma_g)$ by the smallest closed normal subgroup containing the kernels of the two homomorphisms onto $\widehat{F_g}$ , the profinite completion of the free group $F_g$ of rank $g$ , obtained after gluing by a Haar-random element of the profinite completion of $\mathrm{Mod}(\Sigma_g)$ , Haar measure being the canonical translation-invariant probability measure on a profinite (compact, totally disconnected) group, pushed forward to double cosets of the closure of the handlebody subgroup. The question is whether this group almost surely has a finite-index subgroup with a continuous epimorphism onto the profinite integers $\widehat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ , which is the natural profinite formulation of virtual positive Betti number; the authors note that it involves a different order of taking limits than the classical questions, and is heuristically, though not logically, connected to them. Experiments in the source paper and in the authors' earlier census study, where about 1.7 percent of genus-2 hyperbolic manifolds have a $\mathbb{Z}/2$ -cover with $\beta_1 > 0$ , are consistent with all fixed- $Q$ probabilities being 0, but the maximal abelian cover question was stated to be unresolved even experimentally.

Problem Statement

Work in the Dunfield-Thurston random Heegaard splitting model: fix $g \ge 2$ and a finite symmetric generating set of the mapping class group $\mathrm{Mod}(\Sigma_g)$ , let $\varphi$ be the endpoint of a random walk of length $L$ , and let $M_{g,L}$ be the closed 3-manifold obtained by gluing two genus- $g$ handlebodies along $\Sigma_g$ by $\varphi$ . Limits are taken as $L \to \infty$ at fixed $g$ (they are independent of the walk by Dunfield-Thurston); where a part is posed in the large-genus regime, the limit is $L \to \infty$ and then $g \to \infty$ , as in the source. Determine which finite covers of the random manifold have positive first Betti number $\beta_1$ :

(1) For a fixed finite group $Q$ , the open case being $Q$ not solvable (for example $Q = A_5$ , the alternating group on five letters): is the probability that $M_{g,L}$ has a $Q$ -cover, i.e. a regular cover with deck group $Q$ , with $\beta_1 > 0$ , equal to 0? If not, compute the limiting probability. (Known: 0 for every abelian $Q$ at genus 2 by the source paper's Theorem 9.1, for every solvable $Q$ at every fixed genus by Rivin, and for covers of any fixed degree $\le n$ in the large-genus regime by Sawin-Wood.)

(2) Source Question 9.2: what is the probability that the maximal abelian cover of $M_{g,L}$ , that is the cover with deck group $H_1(M_{g,L};\mathbb{Z})$ , typically finite but of expected order growing with $L$ , has $\beta_1 > 0$ ? In particular, is this probability 0, and how does it depend on the genus $g$ ?

(3) Source Question 4.8 (the profinite analogue): for a fixed genus $g \ge 2$ , let $G$ be the profinitefold group defined by a Haar-random profinite Heegaard diagram, i.e. the quotient of the profinite completion of $\pi_1(\Sigma_g)$ by the smallest closed normal subgroup containing the kernels of the two homomorphisms to the profinite free group $\widehat{F_g}$ after gluing by a Haar-random element of the profinite completion of $\mathrm{Mod}(\Sigma_g)$ , with the measure pushed forward to the double cosets of the closure of the handlebody subgroup. Does $G$ have, for almost every such diagram, a subgroup of finite index admitting a continuous epimorphism onto the profinite integers $\widehat{\mathbb{Z}}$ ?

These three parts are one problem: completing the determination, begun in Section 9 of the source paper and advanced by the later work cited above, of the virtual positive-Betti behavior of the finite covers that a random Heegaard splitting actually has, in the regimes where the covering group is fixed, is the maximal abelian one, or is profinite.

References

Nathan M. Dunfield and William P. Thurston, "Finite covers of random 3-manifolds", Inventiones mathematicae 166 (2006), no. 3, 457-521. DOI: 10.1007/s00222-006-0001-6, https://doi.org/10.1007/s00222-006-0001-6 (arXiv:math/0502567, https://arxiv.org/abs/math/0502567).
Nathan M. Dunfield and William P. Thurston, "The virtual Haken conjecture: Experiments and examples", Geometry & Topology 7 (2003), 399-441. DOI: 10.2140/gt.2003.7.399, https://doi.org/10.2140/gt.2003.7.399 (arXiv:math/0209214, https://arxiv.org/abs/math/0209214)
Joseph Maher, "Random Heegaard splittings", Journal of Topology 3 (2010), no. 4, 997-1025. DOI: 10.1112/jtopol/jtq031, https://doi.org/10.1112/jtopol/jtq031 (arXiv:0809.4881, https://arxiv.org/abs/0809.4881)
Gabriele Viaggi, "Volumes of random 3-manifolds", Journal of Topology 14 (2021), no. 2, 504-537. DOI: 10.1112/topo.12190, https://doi.org/10.1112/topo.12190 (arXiv:1905.04935, https://arxiv.org/abs/1905.04935)
Igor Rivin, "Statistics of random 3-manifolds occasionally fibering over the circle", preprint (2014). arXiv:1401.5736, https://arxiv.org/abs/1401.5736
Will Sawin and Melanie Matchett Wood, "Finite quotients of 3-manifold groups", Inventiones mathematicae 237 (2024), no. 1, 349-440. DOI: 10.1007/s00222-024-01262-4, https://doi.org/10.1007/s00222-024-01262-4 (arXiv:2203.01140, https://arxiv.org/abs/2203.01140)
Ian Agol, "The virtual Haken conjecture (with an appendix by Ian Agol, Daniel Groves and Jason Manning)", Documenta Mathematica 18 (2013), 1045-1087. DOI: 10.4171/dm/421, https://doi.org/10.4171/dm/421
Peter Feller, Alessandro Sisto and Gabriele Viaggi, "Uniform models and short curves for random 3-manifolds", Compositio Mathematica 161 (2025), no. 3, 447-502. DOI: 10.1112/S0010437X24007565, https://doi.org/10.1112/S0010437X24007565 (arXiv:1910.09486, https://arxiv.org/abs/1910.09486)
Brent Everitt, "Alternating quotients of Fuchsian groups", Journal of Algebra 223 (2000), no. 2, 457-476. DOI: 10.1006/jabr.1999.8014, https://doi.org/10.1006/jabr.1999.8014 (arXiv:math/9907148, https://arxiv.org/abs/math/9907148)
Hyungryul Baik, David Bauer, Ilya Gekhtman, Ursula Hamenstädt, Sebastian Hensel, Thorben Kastenholz, Bram Petri and Daniel Valenzuela, "Exponential torsion growth for random 3-manifolds", International Mathematics Research Notices 2018 (2018), no. 21, 6497-6534. DOI: 10.1093/imrn/rnx076, https://doi.org/10.1093/imrn/rnx076 (arXiv:1607.00631, https://arxiv.org/abs/1607.00631)