Hyperbolic manifolds that fiber over $S^1$ in higher odd dimensions: existence, abundance, virtual fibration, and structure of the fibrations

The first — and so far only — hyperbolic manifolds known to fiber over the circle in dimension greater than 3 are the cusped, finite-volume hyperbolic 5-manifolds constructed by Italiano, Martelli, and Migliorini in 2021. Their paper closes with a coherent cluster of open problems extending Thurston's fibering theory of hyperbolic 3-manifolds to higher odd dimensions: whether fibering hyperbolic manifolds exist in dimensions $7,9,\ldots$; how many commensurability classes of fibering hyperbolic 5-manifolds there are, and whether compact examples exist in any dimension; whether every odd-dimensional finite-volume hyperbolic manifold is virtually fibred; whether the cone of fibering cohomology classes is polytopal as it is in dimension 3; what the monodromy of the known fibrations is (partially answered in November 2025 by Martelli's construction of an explicit 4-dimensional pseudo-Anosov monodromy); and what geometry the resulting geometrically infinite $F\times\mathbb{R}$ covers carry. Apart from the monodromy advance and the extension of perfect circle-valued Morse functions to dimension 6, all of these components remain open.

Background

A hyperbolic $n$-manifold is a complete Riemannian manifold of constant sectional curvature $-1$; equivalently, a quotient of hyperbolic $n$-space by a discrete group of isometries acting freely. A manifold $M$ fibers over the circle when it is diffeomorphic to the mapping torus of a self-diffeomorphism $\varphi:F\to F$, that is $M\cong(F\times[0,1])/(x,1)\sim(\varphi(x),0)$; the compact manifold $F$ is the fiber and $\varphi$, well defined up to isotopy, is the monodromy. Equivalently, $M$ admits a smooth submersion onto $S^1$.

In dimension 3 the theory is complete and tightly tied to hyperbolic geometry. Thurston proved that a compact 3-manifold fibering over $S^1$ with fiber a surface of negative Euler characteristic admits a hyperbolic metric precisely when the monodromy is pseudo-Anosov — a homeomorphism that, away from a finite singular set, preserves a pair of transverse measured foliations, stretching the leaves of one by a factor $\lambda>1$ and contracting those of the other by $1/\lambda$. The cohomology classes in $H^1(M;\mathbb{R})$ represented by fibrations form an open cone which in dimension 3 is polytopal: it is the cone on the open faces of a polytope (the unit ball of the Thurston norm on cohomology), as the source paper records. Thurston also conjectured that every closed hyperbolic 3-manifold has a finite cover that fibers; this virtual fibration conjecture was proved by Agol in 2013 as a consequence of the virtual specialness of cubulated hyperbolic groups.

Above dimension 3 the picture was, until recently, empty. A mapping torus has Euler characteristic $0$, while every finite-volume hyperbolic manifold of even dimension has nonzero Euler characteristic, so no even-dimensional finite-volume hyperbolic manifold can fiber — or even virtually fiber (Kielak's survey states this as a proposition). Odd dimensions $\geq 5$ were entirely open until Italiano, Martelli, and Migliorini (2021, published 2023) constructed the first examples: finite-volume cusped hyperbolic 5-manifolds fibering over $S^1$ (cusped means non-compact of finite volume, with ends that are quotients of horoballs), including the smallest known hyperbolic 5-manifold, discovered earlier by Ratcliffe and Tschantz. Their method glues copies of a right-angled hyperbolic polytope — a polytope all of whose dihedral angles are $\pi/2$, so that reflections in its facets generate a discrete Coxeter group and gluings of copies along facets can produce complete hyperbolic manifolds — and certifies the fibration by Bestvina–Brady Morse theory, a combinatorial Morse theory on the cubical complex dual to the gluing in which finiteness properties of the kernel of a homomorphism $\pi_1(M)\to\mathbb{Z}$ are read off from the topology of sublevel sets, combined with the criterion of Jankiewicz, Norin, and Wise for right-angled Coxeter groups. Two hyperbolic manifolds are commensurable when they admit a common finite cover; all known fibering 5-manifolds lie in a single commensurability class. A weaker, purely group-theoretic notion also plays a role: a group algebraically fibers if it surjects onto $\mathbb{Z}$ with finitely generated kernel; algebraic fibering does not by itself produce a fibration of the manifold. A circle-valued Morse function is perfect when its number of critical points equals the absolute value of the Euler characteristic; for odd-dimensional manifolds, the existence of such a function with all critical points of middle index is equivalent to fibering, and Kielak calls this the Singer condition.

Finally, a hyperbolic manifold is geometrically infinite when its convex core (the smallest convex submanifold carrying the fundamental group) has infinite volume. The infinite cyclic cover of a fibering hyperbolic 5-manifold is diffeomorphic to $F\times\mathbb{R}$ and is geometrically infinite; in dimension 3 such manifolds are classified by ending invariants (the ending lamination theorem), but nothing analogous is known in dimension 5, where these covers are the first examples available.

Problem Statement

Extend the fibering theory of hyperbolic 3-manifolds to higher odd dimensions. The source paper poses the following linked open questions, which — except where noted — remain open:

1. Existence in higher odd dimensions: do there exist hyperbolic manifolds that fiber over $S^1$ in dimensions $7,9,\ldots$? More generally, do finite-volume hyperbolic manifolds admitting perfect circle-valued Morse functions exist in every dimension (known through dimension 6 only)?

2. Abundance and compactness in dimension 5: how many commensurability classes of hyperbolic 5-manifolds fibering over $S^1$ are there (only one is known)? Do there exist compact (closed) examples? By the Euler-characteristic obstruction recalled in the background, in dimension greater than 3 such examples could only occur in odd dimension at least 5.

3. Virtual fibration in odd dimensions: is every finite-volume hyperbolic manifold of odd dimension finitely covered by one that fibers over $S^1$? Equivalently in the formulation of Battista–Martelli recalled by the source paper, is every hyperbolic manifold finitely covered by one admitting a perfect circle-valued Morse function? No counterexample is known even among higher-dimensional manifolds.

4. Fibering cone: for a fibering hyperbolic 5-manifold $M$ whose first Betti number $b_1(M)=\dim_{\mathbb{R}}H^1(M;\mathbb{R})$ is at least 2, describe the open cone $U\subset H^1(M;\mathbb{R})$ of cohomology classes represented by fibrations. Is $U$ polytopal, as in dimension 3? Is $U$ dense in $H^1(M;\mathbb{R})$, and are there examples on both sides? Compute $U$ for explicit examples or their finite covers.

5. Monodromy and dynamics: determine the monodromy diffeomorphism of explicit fibrations and study its dynamics. This is partially resolved: one example (in the known commensurability class) now has an explicit 4-dimensional pseudo-Anosov monodromy (Martelli, 2025). The monodromy of the original smallest example (the Ratcliffe–Tschantz manifold), the behavior under perturbing the fibration on covers, and a higher-dimensional analogue of Thurston's theorem — whether pseudo-Anosov-type monodromy characterizes hyperbolic mapping tori in general dimension — remain open.

6. Geometry of the infinite cover: for geometrically infinite hyperbolic 5-manifolds diffeomorphic to $F\times\mathbb{R}$, are they rigid? Can a reasonable ending-invariant theory be developed in dimension 5? Do such manifolds exist that do not cover any finite-volume manifold?

These components share one scientific objective: to determine which hyperbolic manifolds in odd dimensions at least 5 fiber — actually or virtually — over the circle, and what structure their fibrations and associated infinite covers carry. Each component is meaningful relative to this common objective, and progress on any of them (a new commensurability class, a dimension-7 example, a computation of $U$, an ending-invariant construction) is a direct contribution to it. The six components are independently asked.

References

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