CARTAN ACTIONS OF HIGHER RANK ABELIAN GROUPS AND THEIR CLASSIFICATION RALF SPATZIER ∗ AND KURT VINHAGE y k ` Abstract. We study transitive R ×Z actions on arbitrary compact manifolds with a projectively dense set of Anosov elements and 1-dimensional coarse Lyapunov foliations. Such actions are called totally Cartan actions. We completely classify such actions as built from low-dimensional Anosov flows and diffeomorphisms and affine actions, verifying the Katok-Spatzier conjecture for this class. This is achieved by introducing a new tool, the action of a dynamically defined topological group describing paths in coarse Lyapunov foliations, and understanding its generators and relations. We obtain applications to the Zimmer program. Contents Introduction and Main Results 2 1. Introduction 2 2. Summary of Arguments7 Preliminaries, Examples and Standard Structure Theory 13 3. Dynamical Preliminaries 13 4. Topological group preliminaries 19 5. Lyapunov Metrics and Closing Lemmas 22 6. Examples of Totally Cartan Actions and Classification of Affine Actions 34 Part I. Rank One Factors and Transitivity of Hyperplane Actions 41 7. Basic properties of Lyapunov central manifolds 41 8. W H -Holonomy Returns to Lyapunov central manifolds 43 9. Finite returns: Constructing a rank one factor 48 10. Infinite returns: Finding a dense H-orbit 53 11. Proofs of Theorems 2.1 and 2.2 61 Part II. Constructing a Homogeneous Structure 61 arXiv:1901.06559v3 [math.DS] 13 Sep 2020 12. Basic Structures of Geometric Brackets 62 13. Ideals of Weights 70 14. Homogeneity from Pairwise Cycle Structures 78 15. Extensions by Maximal Ideals 86 16. Proofs of Theorems 1.7 and 12.19 97 Part III. Structure of General Totally Cartan Actions 98 17. The Starkov component 98 ∗ Supported in part by NSF grants DMS 1607260 and DMS 2003712. y Supported in part by NSF grant DMS 1604796. 1 18. Centralizer Structure 99 19. Virtually Self-Centralizing Actions 102 20. Applications of the Main Theorems 108 References 110 Introduction and Main Results 1. Introduction Hyperbolic actions of higher rank abelian groups are markedly different from single hyperbolic diffeomorphisms and flows and display a multitude of rigidity properties such as measure rigidity and cocycle rigidity. We refer to the survey [72] as a starting point for these topics, and [75, 16, 18,5] and [47, 12, 77] for more recent developments concerning measure and cocycle rigidity, respectively. In this paper, we concentrate on the third major rigidity property: classification and global differential rigidity of such actions. Smale already conjectured in 1967 that generic diffeomorphisms only commute with its iterates [71]. In the hyperbolic case this was proved by Palis and Yoccoz in 1989 [56, 57]. Recently, Bonatti, Crovisier and Wilkinson proved this in full generality [4]. Much refined local rigidity properties were found in the works of Hurder and then Katok and Lewis on deformation and local rigidity of the standard action of SL(n; Z) on the n-torus in which they used properties of the action of higher rank abelian subgroups [35, 43, 44]. This led to investigating such actions more systematically. Let us first coin some terminology. Given a foliation F, a diffeomorphism a acts normally hyper- bolically w.r.t. F if F is invariant under a and if there exists a C0 splitting of the tangent space s u TM = Ea ⊕ T F ⊕ Ea into stable and unstable subspaces for a and the tangent space of the leaves s u of F. More precisely, we assume that Ea and Ea are contracted uniformly in either forward or backward time by a (see Definition 3.1 for precise definitions). All manifolds discussed will be assumed to be connected unless otherwise stated. This will often be superfluous, since we often assume the existence of a continuous action with a dense orbit. k ` k ` 1 Definition 1.1. Let R × Z , and R × Z y X be a locally free C action on a compact manifold k k ` X. If a acts normally hyperbolic to the orbit foliation of R for some a 2 R × Z , we call a an Anosov element, and α an Anosov action. If k + ` ≥ 2, the action is called higher rank. k ` k+` k+`−1 Let A ⊂ R × Z denote the set of Anosov elements, and p : R n f0g ! RP denote the projection onto real projective space. We say that an action is totally Anosov if p(A) is dense in k+`−1 RP . k ` k ` An action R × Z y X is called transitive is there exists a point x 2 X such that (R × Z ) · x is dense in X. ` Note that higher rank Anosov Z -actions correspond to ` commuting diffeomorphisms with at least one being Anosov. The following definition is critical to the structure theory of higher rank abelian actions. k ` r 1 Definition 1.2. Let k + ` ≥ 2 and R × Z y X be a C action on a C manifold X, r ≥ 1.A Cs-rank one factor, s ≤ r is • a C1 manifold Y and a Cs submersion π : X ! Y , k ` • a homomorphism p : R × Z ! A, where A is a compact extension of R or Z, and s • a C action A y Y such that π(a · x) = p(a) · π(x). 2 The factor is called a Kronecker factor if the action of A is by translations of a torus. k ` Remark 1.3. If R ×Z y X, π : X ! Y is a Kronecker factor, it is not difficult to see that any two elements related by a stable or unstable manifold must be sent to the same point. In particular, ` k there are no Kronecker factors of an Anosov Z action, and the only Kronecker factors of an R action correspond to projecting onto the coordinates of the acting group. Our definition coincides with another one that is often used: one may alternatively define a Cs- 0 k l rank one factor as a factor for which a cocompact subgroup A ⊂ R × Z acts by a single flow or diffeomorphism after projecting (A0 is complementary to the A of Definition 1.2). The lack of a rank one factor can be interpreted as an irreducibility condition: many actions with rank one factors are represented as skew-products in the corresponding category where the base of the skew product is a flow. One may similarly define measurable and continuous (C0) factors of the action, in which the regularity conditions are relaxed. It is clear that every smooth rank one factor is topological, and every topological rank one factor is measurable. Forbidding smooth rank one factors is therefore weakest of the possible no rank-one factor assumptions. There are natural examples of higher rank actions coming from homogeneous actions, e.g. by the diagonal subgroup of SL(n; R) on a compact quotient SL(n; R)=Γ or actions by automorphisms of Lie groups, e.g. by commuting toral automorphisms. More generally, genuinely higher rank actions should be algebraic in the following sense (see Section6 for examples of such Anosov actions): k l Definition 1.4. An algebraic action is an action by R × Z by compositions of translations and automorphisms on a compact homogeneous space G=Γ. The action is homogeneous if it consists only of translations. Katok and Spatzier proved local C1 rigidity of standard actions without rank one factors in [46], generalizing the earlier works by Hurder [35, Theorem 2.19] respectively Katok and Lewis [43, Theorem 4.2]. This gives evidence for the following conjecture of Katok and Spatzier (posed as a question in [9] and as a conjecture in [33, Conjecture 16.8]). Conjecture 1.5. (Katok-Spatzier) All higher rank C1 Anosov actions on any compact manifold without C1 rank one factors are C1 conjugate to an algebraic action after passing to finite covers. A Cr version of the conjecture may also hold, possibly with loss of regularity in the conjugacy. This loss of regularity occurs in many rigidity theorems for partially hyperbolic actions, see, e.g. [12]. The minimal value of r for which the conjecture should be true is still mysterious. Conjecture 1.5 is reminiscent of the longstanding conjecture by Anosov and Smale that Anosov diffeomorphisms are topologically conjugate to an automorphism of an infra-nilmanifold [71]. The higher rank and irreducibility assumptions allow for much more dramatic conclusions: First, the conjugacy is claimed to be smooth. For single Anosov diffeomorphisms, smooth rigidity results are not possible as one can change derivatives at fixed points by local changes. Furthermore, Farrell and Jones and later Farrell and Gogolev constructed Anosov diffeomorphisms on exotic tori [20, 19, 21] (although after passing to a finite cover, the exotic structure becomes standard). Thus not even the differentiable structure of the underlying manifold is determined. Secondly, the conjecture applies k k equally well to Z and R -actions. There is no version of the Anosov-Smale conjecture for Anosov flows. Even topological rigidity is out of reach as there are many flows which are not even orbit- equivalent to Anosov flows, see for instance the Handel-Thurston examples [31]. In addition, Anosov flows may not be transitive, such as the Franks-Williams examples [25]. Significant progress has been made on this conjecture in the last decade. For higher rank Anosov k Z actions on tori and nilmanifolds, Rodriguez Hertz and Wang [66] have proved the ultimate result: 3 global rigidity assuming only one Anosov element, and that its linearization does not have affine rank one factors. This improves on the results of [64], and together they are the only known results that assume the existence of only one Anosov element in the action.