The compact-group Chebotarev law for knots: classification beyond semisimple targets

McMullen (2013) proved that the closed orbits of any topologically mixing pseudo-Anosov flow on a closed 3-manifold obey a Chebotarev law, making sequences of knots behave like prime numbers: for any finite group $G$ and any surjective homomorphism $\rho$ from the fundamental group of the complement of the first $n$ knots onto $G$, the conjugacy classes of the remaining knots equidistribute over the conjugacy classes of $G$ with the expected densities. He also established a compact-group version — for a compact Lie group $G$ and a homomorphism $\rho$ with dense image, the holonomy classes of the knots equidistribute with respect to Haar measure — but only under the hypothesis that the identity component $G^0$ is semisimple, and he gave an explicit example showing the statement fails for $G=S^1$. No published work determines necessary and sufficient conditions on the flow and the representation $\rho$ under which the compact-group Chebotarev law holds when $G^0$ is not semisimple, nor what limiting distribution replaces Haar equidistribution in the failing cases (in the known counterexample the holonomies concentrate near a value determined by the period). The problem asks for this classification, together with the replacement statement for the failing cases, as the completion of the compact-group Chebotarev law for knots.

Background

In number theory, the Chebotarev density theorem describes how prime ideals distribute among conjugacy classes. Let $L/K$ be a finite Galois extension of number fields with Galois group $G$ (the group of field automorphisms of $L$ fixing $K$). To every prime ideal of $K$ unramified in $L$ one attaches a Frobenius conjugacy class in $G$, which encodes the arithmetic action of the prime on the residue field extension. Chebotarev's theorem asserts that the natural density (the limit of the counting proportion as the norm bound tends to infinity) of primes whose Frobenius class is a given class $C$ exists and equals $|C|/|G|$; it contains the prime number theorem as the case of the trivial group.

Arithmetic topology is a dictionary, developed by Mazur and Morishita, between 3-dimensional topology and number theory: closed 3-manifolds correspond to number rings, knots to prime ideals, and the fundamental group of a knot complement (the knot group) to a Galois group. In this dictionary the Chebotarev theorem should have a topological counterpart. Mazur asked whether Chebotarev arrangements of knots exist, and McMullen gave an affirmative answer of dynamical origin. Following his formulation, a sequence of disjoint, smooth, oriented knots $(K_i)_{i\ge1}$ in a closed, connected 3-manifold $M$ obeys the Chebotarev law if, writing $L_n=K_1\cup\dots\cup K_n$, for every finite group $G$, every surjective homomorphism $\rho:\pi_1(M-L_n)\to G$, and every conjugacy class $C\subset G$,

$$\lim_{N\to\infty}\frac{|\{n<i\le N : [K_i]=C\}|}{N}=\frac{|C|}{|G|},$$

where $[K_i]\subset G$ is the conjugacy class obtained as follows: a push-off of $K_i$ into the complement determines an element of $\pi_1(M-L_n)$, well-defined up to conjugacy, and $[K_i]$ is its image under $\rho$.

McMullen proved: (Theorem 1.1) if $X$ is a closed surface of constant negative curvature and $M=T_1(X)$ is its unit tangent bundle (the 3-manifold of unit tangent vectors of $X$, carrying the geodesic flow, i.e., the flow moving each unit vector along the geodesic it determines), then the closed orbits of the geodesic flow — each an embedded knot in $T_1(X)$ — ordered by length, obey the Chebotarev law. (Theorem 1.2) the same holds for the closed orbits of any topologically mixing pseudo-Anosov flow on a closed 3-manifold. A pseudo-Anosov flow is a flow preserving a pair of transverse singular foliations, uniformly expanding along one and contracting along the other; the model examples are geodesic flows on negatively curved surfaces and suspensions of pseudo-Anosov surface homeomorphisms (mapping classes with this foliation behavior). Topological mixing means that any two nonempty open sets are moved into contact by the flow at all sufficiently large times; for these flows it is equivalent to the existence of two closed orbits whose lengths have irrational ratio. Via Dehn surgery — removing a tubular neighborhood of a knot and regluing it by a boundary homeomorphism — the law transfers to $S^3$: the knots arising from the periodic cycles of monodromy of the figure-eight knot's fibration, and more generally of any fibered hyperbolic knot in $S^3$, obey the Chebotarev law (Corollary 1.3). A special case of Theorem 1.1, for covers of $T_1(X)$ induced by covers of $X$, had been obtained earlier by Sunada.

The compact-group refinement is the subject of this problem. Let $G$ be a compact Lie group and $\rho:\pi_1(M-L_n)\to G$ a homomorphism with dense image. Each knot $K_i$ with $i>n$ again determines a holonomy conjugacy class $[K_i]\subset G$, and the compact-group Chebotarev law asserts

$$\frac{1}{N}\sum_{i=n+1}^{N} f([K_i]) \to \int_G f\,d\mathrm{Haar}$$

for every continuous class function $f$ on $G$ (a function constant on conjugacy classes), where $\mathrm{Haar}$ is the unique bi-invariant probability (Haar) measure on $G$. McMullen's Theorem 1.4 establishes this whenever the identity component $G^0$ is semisimple — that is, $G^0$ has no nontrivial connected abelian quotient, equivalently its Lie algebra is a direct sum of simple ideals; the groups $SU(2)$ and $SO(3)$ qualify, while the circle $S^1$ does not. For example, for $G=SU(2)$ the normalized traces of the holonomies follow the Sato–Tate measure $(2/\pi)\sqrt{1-x^2}\,dx$ on $[-1,1]$, the same measure as in the Sato–Tate conjecture for elliptic curves over $\mathbb{Q}$ without complex multiplication. However, Theorem 1.4 can fail when $G=S^1$: in section 6 of the same paper, McMullen constructs a topologically mixing pseudo-Anosov flow (the suspension of a pseudo-Anosov surface map acting trivially on homology, rescaled by a closed 1-form) and a dense homomorphism $\rho:\pi_1(M)\to S^1$ for which the holonomies $L(K)\bmod 1$ of orbits with $L(K)\le T$ concentrate near the single value $T\bmod 1$ rather than spreading uniformly — there are exponentially more long orbits than short ones, so each window of periods dominates its predecessors.

The proofs rest on symbolic dynamics. Markov sections (finitely many transversals cutting a hyperbolic flow into rectangles) code the flow as a suspension of a subshift of finite type — a combinatorial system whose states and transition rules reproduce every orbit — with a Hölder continuous roof function (Hölder continuity being a regularity condition between continuity and differentiability) giving return times. Parry and Pollicott proved (their Theorem 8.5) that such a suspension, equipped with a Hölder cocycle into a compact group, obeys the Chebotarev law whenever the resulting compact-group extension has a dense orbit and no circle factor, where a circle factor is a continuous map $p$ from the space to $S^1$ conjugating the flow to a rigid rotation: $p(t\cdot x)=e^{iat}p(x)$. Under the semisimplicity hypothesis, any circle factor of the extension would induce a nontrivial continuous homomorphism $G\to S^1$, which semisimple $G^0$ forbids; for tori and other non-semisimple targets, characters survive, and exactly this mechanism produces the counterexample. The known picture thus stops at two endpoints: a sufficient condition (semisimple) and a single counterexample, with nothing published in between.

The quantitative side of the surrounding literature is instructive as context. For closed geodesics, the distribution of periods in homology classes and error terms for holonomy equidistribution have been studied intensively: Sarnak and Wakayama proved equidistribution of holonomy about closed geodesics with an error term, and Pollicott and Sharp obtained error terms for the distribution of holonomies in circle extensions of hyperbolic flows. Coles and Sharp determined precise asymptotics for the number of closed orbits of a transitive Anosov flow (the smooth, uniformly hyperbolic analogue of a pseudo-Anosov flow, expanding along one direction and contracting along another) in given homology classes of a knot complement — the abelian, discrete-target analogue with orbit counts — using McMullen's theorem in their argument. Pollicott and Zhang proved rapid mixing for compact group extensions of hyperbolic flows and error estimates for holonomy equidistribution decaying faster than any polynomial: for semisimple $G$ when the Brin group (the subgroup generated by the holonomies of closed orbits of the extension) is dense, and for any compact connected $G$ including tori under a Diophantine condition on finitely many holonomy values (an arithmetic non-degeneracy condition asserting that certain real numbers admit no too-good rational approximations). Their results require smooth extensions and rapidly mixing base flows, and do not treat pseudo-Anosov flows, knot-complement representations, the failing-$S^1$ dichotomy, or knot-length orderings. What remains open, and what this problem asks for, is the completion of the compact-group Chebotarev law for knots: the classification of when it holds beyond the semisimple case, together with the replacement statement in the failing cases.

Problem Statement

Let $M$ be a closed 3-manifold equipped with a topologically mixing pseudo-Anosov flow $\varphi$, with closed orbits $(K_i)_{i\ge1}$ ordered by length (the standard examples being the geodesic flow on the unit tangent bundle $T_1(X)$ of a closed surface $X$ of constant negative curvature, and the flows transferred to $S^3$ by Dehn surgery in McMullen's Corollary 1.3). Let $G$ be a compact Lie group, let $L_n=K_1\cup\dots\cup K_n$, and let $\rho:\pi_1(M-L_n)\to G$ be a homomorphism with dense image; each later orbit determines a holonomy conjugacy class $[K_i]\subset G$. The compact-group Chebotarev law asserts that

$$\frac{1}{N}\sum_{i=n+1}^{N} f([K_i]) \to \int_G f\,d\mathrm{Haar}$$

for every continuous class function $f$ on $G$. It is known to hold when the identity component $G^0$ is semisimple, and to fail for an explicit mixing pseudo-Anosov flow with $G=S^1$, where the holonomies of orbits of length at most $T$ concentrate near $T\bmod 1$.

The problem, as the completion of the compact-group Chebotarev law for knots:

(1) Classification beyond the semisimple case. For all compact $G$ with $G^0$ not semisimple, determine necessary and sufficient conditions on the triple $(M,\varphi,\rho)$ under which the compact-group Chebotarev law holds.

(2) Replacement statement in the failing cases. Where the law fails, determine whether the normalized counting measures of the holonomy classes $[K_i]$ converge and, where a limit exists, identify the limiting distribution; where no limit exists, characterize the behaviour that replaces Haar equidistribution.

References

1. Curtis T. McMullen, "Knots which behave like the prime numbers", Compositio Mathematica 149(8) (2013) 1235-1244. DOI: 10.1112/S0010437X13007173. URL: https://doi.org/10.1112/S0010437X13007173.
2. William Parry and Mark Pollicott, "Zeta functions and the periodic orbit structure of hyperbolic dynamics", Astérisque 187-188, Société Mathématique de France, Paris, 1990, 268 pp. Zbl 0726.58003. URL: https://zbmath.org/?q=an:0726.58003.
3. Toshikazu Sunada, "Geodesic flows and geodesic random walks", in Geometry of Geodesics and Related Topics (Tokyo, 1982), Advanced Studies in Pure Mathematics 3, North-Holland, Amsterdam, 1984, pp. 47-85. DOI: 10.2969/aspm/00310047. URL: https://doi.org/10.2969/aspm/00310047. Zbl 0599.58037.
4. Barry Mazur, "Finding meaning in error terms", Bulletin of the American Mathematical Society 45(2) (2008) 185-228. DOI: 10.1090/S0273-0979-08-01207-x. URL: https://doi.org/10.1090/S0273-0979-08-01207-x.
5. Barry Mazur, "Primes, Knots and Po", in Essays on Topology, Springer, 2025, pp. 13-31. DOI: 10.1007/978-3-031-81414-3_2. URL: https://doi.org/10.1007/978-3-031-81414-3_2.
6. Jun Ueki, "Chebotarev links are stably generic", Bulletin of the London Mathematical Society 53(1) (2021) 82-91. DOI: 10.1112/blms.12400. URL: https://doi.org/10.1112/blms.12400. Also arXiv:1902.06906.
7. Jun Ueki, "Modular knots obey the Chebotarev law", preprint, arXiv:2105.10745 (v5, December 2024). URL: https://arxiv.org/abs/2105.10745.
8. Solly Coles and Richard Sharp, "Distribution of periodic orbits in the homology group of a knot complement", Geometriae Dedicata 218(3) (2024), article 75. DOI: 10.1007/s10711-024-00927-x. URL: https://doi.org/10.1007/s10711-024-00927-x. Also arXiv:2207.13608.
9. Mark Pollicott and Daofei Zhang, "Rapid mixing for compact group extensions of hyperbolic flows", Transactions of the American Mathematical Society 378(7) (2025) 5011-5056. DOI: 10.1090/tran/9424. URL: https://doi.org/10.1090/tran/9424. Also arXiv:2405.07356.
10. Mark Pollicott and Richard Sharp, "Periodic orbits and holonomy for hyperbolic flows", in Geometric and Probabilistic Structures in Dynamics, Contemporary Mathematics 469, American Mathematical Society, Providence, RI, 2008, pp. 289-302. Chapter DOI: 10.1090/conm/469/09172 (book DOI: 10.1090/conm/469). URL: https://doi.org/10.1090/conm/469/09172.
11. Peter Sarnak and Masato Wakayama, "Equidistribution of holonomy about closed geodesics", Duke Mathematical Journal 100(1) (1999) 1-57. DOI: 10.1215/s0012-7094-99-10001-9. URL: https://doi.org/10.1215/s0012-7094-99-10001-9. Zbl 0939.22008.
12. Toshiki Matsusaka and Jun Ueki, "Modular knots, automorphic forms, and the Rademacher symbols for triangle groups", Research in the Mathematical Sciences 10 (2023), article 4. DOI: 10.1007/s40687-022-00366-8. URL: https://doi.org/10.1007/s40687-022-00366-8. Also arXiv:2109.01114.
13. Hirofumi Niibo and Jun Ueki, "A Hilbert reciprocity law on 3-manifolds", Research in the Mathematical Sciences 10 (2022), article 3. DOI: 10.1007/s40687-022-00364-w. URL: https://doi.org/10.1007/s40687-022-00364-w.
14. Mark Pollicott, "Dynamical zeta functions and the distribution of orbits", in Handbook of Group Actions V (Lizhen Ji et al., eds.), Advanced Lectures in Mathematics 48, International Press, Somerville, MA / Higher Education Press, Beijing, 2020, pp. 399-440. Zbl 1522.37038. URL: https://zbmath.org/?q=an:1522.37038.
15. Daofei Zhang, "Rapid mixing and superpolynomial equidistribution for torus extensions of hyperbolic flows", preprint, arXiv:2405.11367 (v3, November 2025); to appear in Nonlinearity. URL: https://arxiv.org/abs/2405.11367. Companion to Pollicott-Zhang 2025, treating torus targets under an inhomogeneous Diophantine condition.