Smooth and analytic actions of $SL(n,\mathbf{R})$ and $SL(n,\mathbf{Z})$ on closed $n$-dimensional manifolds

Abstract

The main result 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 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 results about invariant rigid geometric structures for $SL(n,\mathbf{R})$-actions.

arXiv comment29 pp.; In memory of Fuichi Uchida (1938--2021). Minor changes in second version