Smooth classification of Cartan actions of higher rank semisimple Lie groups and their lattices

Edward R. Goetze and Ralf J. Spatzier

Review from Zentralblatt MATH:

Let $G$ be a connected semisimple Lie group without compact factors, whose real rank is at least 2, and let $\Gamma\subset G$ be an irreducible lattice. The authors study the rigidity properties related to the volume preserving actions of a maximal abelian subgroup $A$ of $G$ on a compact manifold $M$. They show that there exists a Hölder-Riemann metric on the studied manifold with respect to which $A$ has uniform expansion and contraction. Then they study the regularity of this metric and of various unions of stable and unstable foliations. The obtained results allow a classification of volume preserving Cartan actions of $\Gamma$ and $G$. If $G$ has real rank at least 3, the authors provide a $C^\infty$ classification for volume preserving, multiplicity free, trellised Anosov actions on compact manifolds. Theorem 1.2. Let $G$ be a connected semisimple Lie group without compact factors and with real rank at least three, and let $A\subset G$ be a maximal $R$-split Cartan subgroup. Let $M$ be a compact manifold without boundary, and let $\mu$ be a smooth volume form on $M$. If $\rho: G\times M\to M$ is an Anosov action on $M$ which preserves $\mu$, is multiplicity free, and is trellised with respect to $A$, then, by possibly passing to a finite cover of $M,\rho$ is $C^\infty$ conjugate to an affine algebraic action, i.e., there exist (1) a finite cover $M'\to M$, (2) a connected, simply connected Lie group $L$, (3) a cocompact lattice $\Lambda\subset L$, (4) a $C^\infty$ diffeomorphism $\varphi: M\to L/ \Lambda$, and (5) a homomorphism $\sigma:G\to Aff(L/ \Lambda)$ such that $\rho'(g)= \varphi^{-1} \sigma(g) \varphi$, where $\rho'$ denotes the lift of $\rho$ to $M'$. Theorem 1.5. Let $G$ be a connected semisimple Lie group without compact factors such that each simple factor has real rank at least 2, and let $\Gamma\subset G$ be a lattice. Let $M$ be a compact manifold without boundary and $\mu$ a smooth volume form on $M$. Let $\rho: \Gamma\times M\to M$ be a volume preserving Cartan action. Then, on a subgroup of finite index, $\rho$ is $C^\infty$ conjugate to an affine algebraic action.

Reviewed by V.Oproiu

Keywords: semisimple Lie group; irreducible lattice; rigidity properties; volume preserving actions; Cartan actions; Anosov actions

Classification (MSC2000): 22E46 22E40

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