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

Let g, p ∈ [0↑∞ [, the set of non-negative integers. Let Ag,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 Π j∈[p↓1] tj Π 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 Ag,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 Ag,p; we do not know an algebraic proof of this for g > 2. Consider an orientable surface Sg,p of genus g with p punctures, with (g, p) 6= (0, 0), (0, 1). The algebraic mapping-class group of Sg,p, denoted M g,p, is defined as the group of all those outer automorphisms of 〈 t[1↑p] ∪ x[1↑g] ∪ y[1↑g] | Π j∈[p↓1] tj Π i∈[1↑g] [xi, yi] 〉 which permute the set of conjugacy classes { [tj ], [tj ] : j ∈ [1↑p]}. 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 Sg,p, along lines suggested by Magnus, Karrass, and Solitar in 1966. 2010 Mathematics Subject Classification. Primary: 20E05; Secondary: 20E36, 20F05, 57M60, 57M05.