Abelian and solvable subgroups of the mapping class group
type: Item
instance of: scholarly article
title: Abelian and solvable subgroups of the mapping class group
zbMATH Open document ID: 0551.57004
DOI: 10.1215/S0012-7094-83-05046-9
author: Joan Birman
author: Alexander Lubotzky
author: John D. McCarthy
published in: Duke Mathematical Journal
publication date: 1983-01-01T00:00:00Z
review text: The main result of the paper is the following theorem: every solvable subgroup of the mapping class group of an orientable surface contains an abelian subgroup of finite index. This result is complemented by bounds on the rank and the index of this subgroup; the first one is precise. The proofs are based on Thurston's theory of diffeomorphisms of surfaces, to which the authors give some nice complements. The importance of the paper lies in that it indicates the effectiveness of Thurston's theory in the study of algebraic properties of the mapping class groups of surfaces. The paper has become the starting point of some further works. In particular, the above theorem was generalized by J. McCarthy and the reviewer (simultaneously and independently) to the following analogue of a famous Tits' theorem in the theory of linear groups: every subgroup of the mapping class group of an orientable surface contains either a free group with two generators or an abelian subgroup of finite index. See the third author, A ``Tits alternative'' for subgroups of surface mapping class groups, Preprint, 1984, and the reviewer, Dokl. Akad. Nauk SSSR 275, 786-789 (1984).
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 57R50
Mathematics Subject Classification ID: 30F99
zbMATH DE Number: 3877980
zbMATH Keywords: free nonabelian subgroups of the mapping class group
zbMATH Keywords: solvable subgroup of the mapping class group of an orientable surface
zbMATH Keywords: abelian subgroup of finite index
zbMATH Keywords: diffeomorphisms of surfaces
MaRDI profile type: Publication
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full work available at URL: s0012-7094-83-05046-9
OpenAlex ID: W1982407034
description: scientific article; zbMATH DE number 3877980