Volume-preserving actions of simple algebraic Q-groups on low-dimensional manifolds

TL;DR

This paper proves that certain high-rank algebraic groups over Q cannot act nontrivially in a volume-preserving way on low-dimensional compact manifolds, extending rigidity results to a broad class of groups.

Contribution

It establishes new rigidity results for volume-preserving actions of simple algebraic Q-groups on low-dimensional manifolds, generalizing previous theorems to a wider class of groups.

Findings

SL(n,Q) has no nontrivial volume-preserving actions on manifolds of dimension less than n

For certain algebraic groups G over Q, actions on low-dimensional manifolds are trivial or factor through finite groups

Conditions on G ensure rigidity of volume-preserving actions in low dimensions

Abstract

We prove that SL(n,Q) has no nontrivial, C-infinity, volume-preserving action on any compact manifold of dimension strictly less than n. More generally, suppose G is a connected, isotropic, almost-simple algebraic group over Q, such that the simple factors of every localization of G have rank at least two. If there does not exist a nontrivial homomorphism from G(R) to GL(d,C), then every C-infinity, volume-preserving action of G(Q) on any compact d-dimensional manifold must factor through a finite group.