Smooth and analytic actions of SL(n,R) and SL(n,Z) on closed n-dimensional manifolds

Abstract

The main theorem is a classification of smooth actions of $SL(n,\mathbf{R})$, $n\ge 3$, or connected groups locally isomorphic to it, on closed n-manifolds, extending a theorem of Uchida. We also construct new exotic actions of $SL(n,\mathbf{Z})$ on the n-torus and connected sums of n-tori, and we formulate a conjectural classification of actions of lattices in $SL(n,\mathbf{R})$ on closed n-manifolds. We prove some related results about invariant rigid geometric structures for $SL(n,\mathbf{R})$-actions.