Anosov-like Group Actions in Hyperbolic Dynamics

• Anosov-like group actions are defined by a smooth group action on manifolds with at least one element inducing uniform hyperbolicity transverse to the orbit foliation.
• They establish an equivalence between analytic, topological, and algebraic properties, linking invariant foliations and solvability of the fundamental group.
• Affine and suspension models reveal rigidity phenomena and drive classification efforts, unifying dynamics, geometry, and algebra in higher-rank settings.

Anosov-like group actions are a framework for describing group actions on manifolds that exhibit robust hyperbolic behavior, closely related to the theory of Anosov flows and diffeomorphisms but encompassing a broader class of group actions, higher-rank abelian and nilpotent actions, and their interconnections with foliations, rigidity, and manifold topology. These actions provide a key setting for unifying dynamical, geometric, and algebraic rigidity phenomena, particularly in the context of actions by abelian, nilpotent, or more general Lie groups, and have played a central role in major classification programs in dynamics and geometry.

1. Definitions and Fundamental Structures

Let $M$ be a closed, connected, orientable manifold of dimension $n+k$, and let $\mathbb{R}^k$ (or a more general Lie group) act smoothly and locally freely on $M$. An action is said to be Anosov-like if there exists at least one group element $a_0 \in \mathbb{R}^k$ such that the induced diffeomorphism $g = \varphi(a_0, \cdot)$ exhibits uniform hyperbolic behavior transverse to the orbit foliation. Specifically, for codimension-one Anosov actions, there is an invariant splitting of the tangent bundle: $TM = E^{ss} \oplus T_{\mathcal{O}} \oplus E^{uu}$ where $T_{\mathcal{O}}$ is tangent to the $\mathbb{R}^k$-orbits, $E^{ss}$ is the strong stable, and $E^{uu}$ is the strong unstable subbundle (with $\dim E^{uu} = 1$ in the codimension-one case) (Barbot et al., 2010).

The dynamical definition requires that for some constants $C > 0$ and $\lambda > 0$,

$$\| Dg^n|_{E^{ss}} \| < C e^{-\lambda n} \qquad \| Dg^{-n}|_{E^{uu}} \| < C e^{-\lambda n} \qquad \forall n > 0.$$

In higher-rank situations, hyperbolicity is typically achieved on cones of group elements rather than a single $(a_0)$, reflecting the richer dynamics of multiparameter actions (Lopes et al., 30 May 2025, Barbot et al., 2012, Barbot et al., 2015).

For non-abelian (e.g., nilpotent) group actions (so-called Nil-Anosov actions), it is required that some Anosov element lies in the nilradical, fundamentally impacting invariance properties of the associated splittings (Barbot et al., 2015).

2. Equivalence of Geometric and Algebraic Properties

A central result for codimension-one $\mathbb{R}^k$-Anosov actions is the equivalence of several analytic, topological, and algebraic features ((Barbot et al., 2010), Theorems 2 and 6):

• (i) The fundamental group $\pi_1(M)$ is solvable.
• (ii) The weak stable foliation $\mathcal{F}^{(s)}$ (tangent to $E^{ss} \oplus T_{\mathcal{O}}$) admits a transverse affine structure of class $C^1$.
• (iii) This transverse affine structure is complete: the developing map $D: M \to \mathbb{R}$ is an affine fibration, with leaves as fibers.
• (iv) The action is splitting: in the universal cover, every strong stable leaf intersects every unstable leaf.

These equivalences tightly link algebraic (solvability) and geometric (foliation, splitting, affine structure) properties, showing that the presence of certain foliation structures forces algebraic constraints on $\pi_1(M)$. This provides higher rank analogues of classical results for Anosov flows, e.g., those due to Ghys for $k=1$.

3. Affine and Algebraic Models: Suspension and Rigidity

Under smoothness and volume-preserving conditions (action $C^2$, $E^{ss} \oplus E^{uu}$ of class $C^1$, invariant volume), Anosov-like actions are topologically equivalent to suspensions of linear Anosov actions of $\mathbb{Z}^k$ on tori: $$M \cong (N \times \mathbb{R})/\mathbb{Z} \qquad \text{with} \qquad (x, t+k) \sim (f^k(x), t)$$ where $f$ is a partially hyperbolic diffeomorphism commuting with an action of $\mathbb{R}^{k-1}$ (Barbot et al., 2010, Barbot et al., 2012). This suspension paradigm unifies the theory of actions of higher rank abelian and nilpotent groups, leading to the result that, up to commensurability, all algebraic Anosov actions are nil-suspensions over suspensions of Anosov $\mathbb{Z}^k$-actions or Weyl chamber actions (the semisimple case) (Barbot et al., 2012).

The presence of rigid geometric structures (generalized contact structures, compatible invariant $k$-geometric forms) further implies that such actions are conjugate to affine (algebraic) actions (Almeida, 2020).

4. Invariant Foliations, Transverse Structures, and Splitting

The invariant splitting grants a family of foliations associated to $E^{ss}$, $E^{uu}$, and their sums with the orbit direction. Key geometric features are:

• Existence and completeness of transverse affine structures on the weak-stable foliation $\mathcal{F}^{(s)}$.
• The completeness condition: developing map $D$ is a fibration over $\mathbb{R}$, fibers are leaves, and holonomy is affine.
• Splitting (in universal cover): every strong stable leaf intersects every unstable leaf; this property is central for topological classification and for the construction of global cross-sections.

These foliations and their transverse structures provide the analytic bridge between algebraic invariants and dynamical properties, leveraging tools such as Tischler's fibration theorem and classification of transverse affine structures.

5. Existence of Cross-Sections and Suspensions

The existence of a global cross-section for a one-parameter subgroup implies (via the Schwartzman criterion: existence of a cohomology class $w \in H^1(M, \mathbb{R})$ with $w(A_\mu) > \epsilon$ for all invariant measures $\mu$) that the action is a "1-suspended" action: the suspension of a partially hyperbolic diffeomorphism whose central direction supports an $(k-1)$-parameter group action commuting with $f$ (Barbot et al., 2010). If the whole action is transverse to a fibration, then after finite covering, $M$ is homeomorphic to the suspension of a $\mathbb{Z}^k$-action.

The dichotomy between minimality of the strong foliations (i.e., denseness of the leaves) and the existence of a fibered (suspension) structure is formalized as a general theorem: for Anosov actions transitive on regular subcones, either all stable and unstable leaves are dense or the manifold is a suspension over a $\mathbb{Z}^k$-Anosov action (Lopes et al., 30 May 2025).

6. Classification and Conjectures

The "generalized Verjovsky conjecture" posits that (irreducible) codimension-one Anosov actions are always topologically equivalent to standard/algebraic suspensions, except possibly for a small set of exceptional cases. This is confirmed in the algebraic context (Barbot et al., 2012), codimension-one nil-suspensions (Barbot et al., 2015), and for broad classes of geometrically rigid actions (Almeida, 2020).

The classification of algebraic Anosov actions of nilpotent Lie groups separates them into nil-suspensions over higher-rank abelian suspensions, modified Weyl chamber actions, or mixed central extensions. The associated Cartan subalgebra (hyperbolic) plays a fundamental invariant role in this classification (Barbot et al., 2012).

Moreover, Anosov group actions (and their representations) have been extensively connected to rigidity and measure-theoretic classification in higher-rank semisimple Lie groups, especially through the paper of Anosov representations, proper actions on homogeneous spaces, and their relations to convex-cocompact subgroups and invariant measures (Guéritaud et al., 2015, Zimmer, 2017, Lee et al., 2020).

7. Broader Implications and Future Directions

The robust connection between the dynamical, algebraic, and geometric structures in Anosov-like group actions supports deep rigidity phenomena; in particular, the transition from minimality of invariant foliations to suspension structures enriches the geometric classification of higher-rank actions and generalizes classical theory for flows and diffeomorphisms. The dichotomy proven for actions transitive on subcones is a decisive step toward confirming the extended Verjovsky conjecture (Lopes et al., 30 May 2025).

Further progress is anticipated in smooth classification (beyond topological conjugacy), a full analysis of cross-section existence, explicit realization of cycles and affine structures in higher-codimension, and extension of rigidity results to more general partially hyperbolic settings. Understanding the interaction of group-theoretic properties (such as solvability, commensurability, and Cartan subalgebras) with the manifold topology and foliation geometry remains a central line of inquiry, with implications for classification problems in dynamical systems, geometry, and higher Teichmüller theory.