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

TL;DR

This paper classifies smooth actions of $SL(n,{f R})$ and related groups on closed $n$-manifolds, introduces new exotic actions of $SL(n,{f Z})$, and discusses invariant geometric structures, extending previous theorems.

Contribution

It extends Uchida's theorem by classifying smooth actions of $SL(n,{f R})$ on closed $n$-manifolds and constructs novel exotic actions of $SL(n,{f Z})$ on tori.

Findings

Classification of smooth $SL(n,{f R})$ actions on closed $n$-manifolds

Construction of new exotic $SL(n,{f Z})$ actions on tori

Results on invariant rigid geometric structures

Abstract

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