Polycyclic groups and transversely affine foliations
type: Item
instance of: scholarly article
title: Polycyclic groups and transversely affine foliations
zbMATH Open document ID: 0767.53021
published in: Journal of Differential Geometry
publication date: 1993-02-18T00:00:00Z
review text: A foliation is called transversely affine if coordinates can be chosen so that the holonomy maps are all affine. It is known that all polycyclic groups occur as the fundamental group of a certain affinely flat manifold. It is conjectured that the converse is virtually true [see \textit{J. Milnor}, Adv. Math. 25, 178-187 (1977; Zbl 0364.55001)]. The author considers here codimension one foliations of manifold with polycyclic fundamental group and shows that in certain natural situations the foliation must have a transverse affine structure. For example, the following theorem is proved. Theorem: Suppose \(\mathcal F\) is a real analytic foliation which has a compact leaf space. If \(\pi_ 1(M)\) is virtually polycyclic and \(\mathcal F\) does not have any invariant measures, then \(\mathcal F\) is minimal and transversely affine. If in addition, \(M\) is compact, then \(\mathcal F\) is covered by a product foliation.
reviewed by: Fabiano G. B. Brito
Mathematics Subject Classification ID: 53C12
Mathematics Subject Classification ID: 53A15
zbMATH DE Number: 120211
zbMATH Keywords: holonomy maps
zbMATH Keywords: polycyclic fundamental group
zbMATH Keywords: virtually polycyclic
author: Joseph F. Plante
MaRDI profile type: Publication
Wikidata QID: Q115181218
full work available at URL: https://doi.org/10.4310/jdg/1214448255
OpenAlex ID: W1537575639
DOI: 10.4310/JDG/1214448255
description: scientific article; zbMATH DE number 120211