These pages are not updated anymore. For the current production of this journal, please refer to http://www.jstor.org/journals/0003486x.html.
Annals of Mathematics, II. Series Vol. 150, No. 3, pp. 743773 (1999) 

Smooth classification of Cartan actions of higher rank semisimple Lie groups and their latticesEdward R. Goetze and Ralf J. SpatzierReview 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 HölderRiemann 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 Full text of the article:
Electronic fulltext finalized on: 8 Sep 2001. This page was last modified: 21 Jan 2002.
© 2001 Johns Hopkins University Press
