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The Zieschang–McCool method for generating algebraic mapping-class groups

The Zieschang–McCool method for generating algebraic mapping-class groups Abstract Let g, p ∈ (0↑∞(, the set of non-negative integers. Let A g,p denote the group consisting of all those automorphisms of the free group on t (1↑ p ) ∪ x (1↑ g ) ∪ y (1↑ g ) which fix the element ∏ j ∈( p ↓1) t j ∏ i ∈(1↑ g ) ( x i , y i ) and permute the set of conjugacy classes {( t j ) : j ∈ (1↑ p )}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A g,p is generated by what is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. (Labruère and Paris also gave defining relations for the ADLH set in A g,p ; we do not know an algebraic proof of this for g ⩾ 2.) Consider an orientable surface S g,p of genus g with p punctures, with ( g, p ) ≠ (0, 0), (0, 1). The algebraic mapping-class group of S g,p , denoted , is defined as the group of all those outer automorphisms of 〈 t (1↑ p ) ∪ x (1↑ g ) ∪ y (1↑ g ) | ∏ j ∈( p ↓1) t j ∏ i ∈(1↑ g ) ( x i , y i )〉 which permute the set of conjugacy classes . It now follows from a result of Nielsen that is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that equals the (topological) mapping-class group of S g,p , along lines suggested by Magnus, Karrass, and Solitar in 1966. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups - Complexity - Cryptology de Gruyter

The Zieschang–McCool method for generating algebraic mapping-class groups

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References (29)

Publisher
de Gruyter
Copyright
Copyright © 2011 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc.2011.007
Publisher site
See Article on Publisher Site

Abstract

Abstract Let g, p ∈ (0↑∞(, the set of non-negative integers. Let A g,p denote the group consisting of all those automorphisms of the free group on t (1↑ p ) ∪ x (1↑ g ) ∪ y (1↑ g ) which fix the element ∏ j ∈( p ↓1) t j ∏ i ∈(1↑ g ) ( x i , y i ) and permute the set of conjugacy classes {( t j ) : j ∈ (1↑ p )}. Labruère and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A g,p is generated by what is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. (Labruère and Paris also gave defining relations for the ADLH set in A g,p ; we do not know an algebraic proof of this for g ⩾ 2.) Consider an orientable surface S g,p of genus g with p punctures, with ( g, p ) ≠ (0, 0), (0, 1). The algebraic mapping-class group of S g,p , denoted , is defined as the group of all those outer automorphisms of 〈 t (1↑ p ) ∪ x (1↑ g ) ∪ y (1↑ g ) | ∏ j ∈( p ↓1) t j ∏ i ∈(1↑ g ) ( x i , y i )〉 which permute the set of conjugacy classes . It now follows from a result of Nielsen that is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that equals the (topological) mapping-class group of S g,p , along lines suggested by Magnus, Karrass, and Solitar in 1966.

Journal

Groups - Complexity - Cryptology – de Gruyter

Published: Dec 1, 2011

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