arXiv subjects

Robert J. Zimmer

Publications and source records attributed to Robert J. Zimmer.

4 recordsLinked to original sources

arXiv2012

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

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.

math.DS↗

arXiv2009

Invariant rigid geometric structures and smooth projective factors

We consider actions of non-compact simple Lie groups preserving an analytic rigid geometric structure of algebraic type on a compact manifold. The structure is not assumed to be unimodular, so an invariant measure may not exist. Ergodic stationary measures always exist, and when such a measure has full support, we show the following. 1) Either the manifold admits a smooth equivariant map onto a homogeneous projective variety, defined on an open dense conull invariant set, or the Lie algebra of the Zariski closure of the Gromov representation of the fundamental group contains a Lie subalgebra isomorphic to the Lie algebra of the acting group. As a corollary, a smooth non-trivial homogeneous projective factor does exist whenever the fundamental group of $M$ admits only virtually solvable linear representations, and thus in particular when $M$ is simply connected, regardless of the real rank. 2) There exist explicit examples showing that analytic rigid actions of certain simple groups (of real rank one) may indeed fail to have a smooth projective factor. 3) It is possible to generalize Gromov's theorem on the algebraic hull of the representation of the fundamental group of the manifold to the case of analytic rigid non-unimodular structures, for actions of simple groups of any real rank. An important ingredient in the proofs is a generalization of Gromov's centralizer theorem beyond the case of invariant measures.

math.DS↗

arXiv2002

Ergodic actions of semisimple Lie groups on compact principal bundles

Let G = SL(n,R) (or, more generally, let G be a connected, noncompact, simple Lie group). For any compact Lie group K, it is easy to find a compact manifold M, such that there is a volume-preserving, connection-preserving, ergodic action of G on some smooth, principal K-bundle P over M. Can M can be chosen independent of K? We show that if M = H/L is a homogeneous space, and the action of G on M is by translations, then P must also be a homogeneous space H'/L'. Consequently, there is a strong restriction on the groups K that can arise over this particular M.

math.DG↗

arXiv2000

Actions of semisimple Lie groups on circle bundles

Suppose G is a connected, simple, real Lie group with real rank at least two, M is an ergodic G-space with invariant probability measure, and f is a Homeo(T)-valued Borel cocycle, where Homeo(T) denotes the group of homeomorphisms of the circle T. We use an argument of E.Ghys to show that there is a G-invariant probability measure on the skew product of M and T. Furthermore, if the image of f consists of diffeomorphisms, then there is an invariant measure that is equivalent to the product measure; therefore, f is cohomologous to a cocycle with values in the isometry group of T.

math.DS↗