Actions of discrete linear groups and Zimmer's conjecture
type: Item
instance of: scholarly article
title: Actions of discrete linear groups and Zimmer's conjecture
zbMATH Open document ID: 0858.22011
author: Renato Feres
published in: Journal of Differential Geometry
publication date: 1996-06-23T00:00:00Z
review text: Let a lattice group \(\Gamma\) of a higher rank semisimple Lie group \(G\) act by bundle automorphisms on a principal \(H\)-bundle \(P\) over a smooth compact manifold \(M\). Assuming that the action of \(\Gamma\) preserves a \(C^r\) connection for some \(r \geq 0\), does not preserve a measurable Riemannian metric and that the smooth measure associated with the \(\Gamma\)-invariant volume form has countably many ergodic components, it is proved that in general this action can be derived from two classical types. More precisely, one has: Either the Zariski closure of the connection's full holonomy group contains a nontrivial homomorphic image of \(G\) or the action is \(C^{r+2}\) isomorphic to: (1) the affine action on flat manifolds and tori (then \(\Gamma\) is a lattice subgroup of \(\mathbb{R}^d\) or a subgroup of \(\text{SL}(d,\mathbb{Z}))\); (2) left translations on compact quotients of non-compact semisimple Lie groups.
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 32N15
zbMATH DE Number: 892614
zbMATH Keywords: lattice subgroups
zbMATH Keywords: volume preserving action
zbMATH Keywords: semisimple Lie group
reviewed by: Jean Lacroix
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214457549
OpenAlex ID: W1494296496
Wikidata QID: Q115177852
DOI: 10.4310/JDG/1214457549
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description: scientific article; zbMATH DE number 892614