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

2012/04/14 by Dave Witte Morris, Morris, Dave Witte, Robert J. Zimmer +1
Mathematics · #20G30 #22F99 #37C85 #57S99 #Dynamical Systems (math.DS) #FOS: Mathematics #math.DS #msc:20G30 #msc:22F99 #msc:37C85 #msc:57S99

paper · pdf · doi:10.48550/arxiv.1204.3139

6 pages, no figures

arxiv created 2012/04/14 · arxiv updated 2012/04/17

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.

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