Math Conjectures

Anosov-manifold conjecture

OPENMajorConjectureProposed c. 1970 · Standard version

Canonical statement

Every closed connected smooth manifold that admits an Anosov diffeomorphism is homeomorphic to an infranilmanifold, namely a quotient Γ\N\Gamma\backslash N, where NN is a simply connected nilpotent Lie group and Γ\Gamma is a torsion-free discrete subgroup of N⋊CN\rtimes C for some compact subgroup C≤Aut⁡(N)C\le\operatorname{Aut}(N), acting freely and cocompactly on NN.
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Every closed connected smooth manifold that admits an Anosov diffeomorphism is homeomorphic to an infranilmanifold, namely a quotient \(\Gamma\backslash N\), where \(N\) is a simply connected nilpotent Lie group and \(\Gamma\) is a torsion-free discrete subgroup of \(N\rtimes C\) for some compact subgroup \(C\le\operatorname{Aut}(N)\), acting freely and cocompactly on \(N\).
DYN-007:[Franks1969AnosovDiffeomorphisms][Manning1974AnosovInfraNil][GogolevLafont2015AsphericalAnosov][DekimpeDerive2019AnosovGraphs]
Dynamical systemsGeometry and topology

Notes

This conjecture seeks a topological classification of the closed manifolds that can carry globally hyperbolic dynamics: if a closed connected smooth manifold admits an Anosov diffeomorphism, it should be homeomorphic to an infranilmanifold Γ\N\Gamma\backslash N, a compact quotient of a simply connected nilpotent Lie group NN. The question crystallized around 1970 out of the work of Franks and Manning on Anosov diffeomorphisms of tori and nilmanifolds [Franks1969AnosovDiffeomorphisms] [Manning1974AnosovInfraNil].

In one direction the picture is complete: Manning showed that Anosov diffeomorphisms of infranilmanifolds are topologically conjugate to hyperbolic affine automorphisms [Manning1974AnosovInfraNil], and every known example of an Anosov diffeomorphism lives on such a manifold. Which infranilmanifolds actually admit them has been analyzed in specific families, for instance those associated to graphs [DekimpeDerive2019AnosovGraphs]. In the other direction, topological and group-theoretic obstructions rule out many candidate manifolds, including various aspherical products [GogolevLafont2015AsphericalAnosov].

These obstructions do not, however, extract an infranil structure from the invariant hyperbolic splitting itself, and no general mechanism for doing so is known. The conjecture remains open.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.