The Zieschang-McCool method for generating algebraic mapping-class groups
Abstract
Let g and p be non-negative integers. Let A(g,p) denote the group consisting of all those automorphisms of the free group on t1,...,tp, x1,...,xg, y1,...yg which fix the element t1t2...tp[x1,y1]...[xg,yg] and permute the set of conjugacy classes [t1],....,[tp]. Labru\`ere and Paris, building on work of Artin, Magnus, Dehn, Nielsen, Lickorish, Zieschang, Birman, Humphries, and others, showed that A(g,p) is generated by a set that is called the ADLH set. We use methods of Zieschang and McCool to give a self-contained, algebraic proof of this result. Labru\`ere 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 > 1. Consider an orientable surface S(g,p) of genus g with p punctures, such that (g,p) is not (0,0) or (0,1). The algebraic mapping-class group of S(g,p), denoted M(g,p), is defined as the group of all those outer automorphisms of the one-relator group with generating set t1,...,tp, x1,...,xg, y1,...yg and relator t1t2...tp[x1,y1]...[xg,yg] which permute the set of conjugacy classes [t1],....,[tp]. It now follows from a result of Nielsen that M(g,p) is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M(g,p) equals the (topological) mapping-class group of S(g,p), along lines suggested by Magnus, Karrass, and Solitar in 1966.