Finite subgroups of mapping class groups of geometric 3-manifolds
type: Item
instance of: scholarly article
title: Finite subgroups of mapping class groups of geometric 3-manifolds
zbMATH Open document ID: 0637.57011
author: Ken'ichi Ohshika
published in: Journal of the Mathematical Society of Japan
publication date: 1987-01-01T00:00:00Z
review text: Let M be a compact Seifert fiber space which fibers over a euclidean or hyperbolic base-2-orbifold. Then M possesses a geometric structure modelled on \({\mathbb{H}}\) \(2\times {\mathbb{R}}\), \(SL_ 2\), \({\mathbb{E}}^ 3 \)or Nil, and the Teichmüller space T(M) of M has been determined in the previously reviewed paper: it is a euclidean space \({\mathbb{R}}^ n.\) As for surfaces, the mapping class group of M acts on T(M), and it is proved in the present paper, using the analogous result for surfaces due to \textit{S. P. Kerckhoff} [Ann. Math., II. Ser. 117, 235-265 (1983; Zbl 0528.57008)], that every finite subgroup G of the mapping class group has a fixed point in T(M). This implies that, under some restrictions, G can be realized by isometries of a geometric structure on M (in general, there is an algebraic obstruction for realizing G, see the paper by the reviewer in Manuscr. Math. 30, 361-373 (1980; Zbl 0438.57005).
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57S17
Mathematics Subject Classification ID: 32G15
Mathematics Subject Classification ID: 57R50
zbMATH DE Number: 4038813
zbMATH Keywords: Seifert fiber space which fibers over a euclidean or hyperbolic base-2- orbifold
zbMATH Keywords: geometric structure
zbMATH Keywords: Teichmüller space
zbMATH Keywords: mapping class group
zbMATH Keywords: finite subgroup G of the mapping class group
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03930447
OpenAlex ID: W2083293339
DOI: 10.2969/JMSJ/03930447
description: scientific article; zbMATH DE number 4038813