Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions
type: Item
instance of: scholarly article
title: Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions
zbMATH Open document ID: 0571.57031
DOI: 10.1016/0166-8641(85)90070-7
author: Yoshinobu Kamishima
published in: Topology and its Applications
publication date: 1985-01-01T00:00:00Z
review text: Let \(\pi\) be a group acting properly discontinuously on a contractible manifold X, such that the action extends to an action of a real algebraic group G containing \(\pi\). If such a \(\pi\) has a nontrivial radical (unique maximal normal solvable subgroup), then it is shown in the present paper that the quotient X/\(\pi\) has the structure of an injective Seifert fiber space, as introduced by Conner-Raymond, with typical resp. exceptional fiber diffeomorphic to a solv- respectively infrasolv- manifold. Moreover if G is an amenable algebraic group, then such an action is essentially unique (up to conjugation by diffeomorphisms). Several applications are given, among them: 1) A complete affinely flat manifold admits a nontrivial (maximal) torus action if its fundamental group has nontrivial center; 2) if the fundamental group of such a manifold is virtually polycyclic, then it has the structure of an infrasolv-manifold; 3) two virtually polycyclic affine crystallographic groups are conjugate in \(Diff({\mathbb{R}}^ n)\) iff they are isomorphic (the last 2 results are due to \textit{D. Fried} and \textit{W. M. Goldman} [Adv. Math. 47, 1-49 (1983; see the preceding review)]).
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 3911324
zbMATH Keywords: properly discontinuous group actions
zbMATH Keywords: injective Seifert fiber space
zbMATH Keywords: infrasolv-manifold
zbMATH Keywords: amenable algebraic group
zbMATH Keywords: complete affinely flat manifold
zbMATH Keywords: torus action
zbMATH Keywords: fundamental group
zbMATH Keywords: polycyclic affine crystallographic groups
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: 0166-8641(85)90070-7
OpenAlex ID: W2052677125
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description: scientific article; zbMATH DE number 3911324