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

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] tj ∏ i∈[1↑g][xi, yi ] and permute the set of conjugacy classes {[tj ] : 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] tj ∏ i∈[1↑g][xi, yi ]〉 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.