Global Rigidity of Higher Rank Anosov Actions on Tori and Nilmanifolds

An Anosov diffeomorphism f on a torus T is affine if f lifts to an affine map on R. By a classical result of Franks and Manning, any Anosov diffeomorphism g on T is topologically conjugate to an affine Anosov diffeomorphism. More precisely, there is a homeomorphism φ : T → T such that f = φ◦g◦φ−1 is an affine Anosov diffeomorphism. We call φ the Franks-Manning conjugacy. The linear part of f is the map induced by g on H1(T). Anosov diffeomorphisms are rarely C-conjugate to affine ones. For example, one can perturb a linear Anosov diffeomorphism locally around a fixed point p to change the conjugacy class of the derivative at p. The resulting diffeomorphism will still be Anosov but cannot be C-conjugate to its linearization. The situation is radically different for Z-actions with many Anosov diffeomorphisms. In other words, Anosov diffeomorphisms rarely commute with other Anosov diffeomorphisms. It follows easily from the result for a single Anosov diffeomorphism that an Anosov Z-action α on T is topologically conjugate to a Z-action by affine Anosov diffeomorphisms. We call this action the linearization of α and denote it by ρ. Again, for any a ∈ Z the linear part of ρ(a) is the map induced by α(a) on H1(T). The logarithms of the moduli of the eigenvalues of these linear parts define additive maps λi : Z → R, which extend to linear functionals on R. A Weyl chamber of ρ is a connected component of R − ⋃ i kerλi. Theorem 1.1. Let α be a C∞-action of Z, k ≥ 2, on a torus T and let ρ be its linearization. Suppose that there is a Z subgroup of Z such that ρ(a) is ergodic for every nonzero a ∈ Z. Further assume that there is an Anosov element for α in each Weyl chamber of ρ. Then α is C∞-conjugate to ρ. Furthermore, for a linear Z-action on T having a Z subgroup acting by ergodic elements is equivalent to several other properties, in particular to being genuinely higher rank [37]. A linear Z-action is called genuinely higher rank if for all finite index subgroups Z of Z, no quotient of the Z-action factors through a finite extension of Z. Hence we obtain the following corollary.