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

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↑] ∪ y [1↑g] which fix the element Π j∈[p↓1]r jΠ i∈[1↑g]][x i, yi] 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 M algg,p is defined as the group of all those outer automorphisms of <t [1↑p] ∪x [1↑g] | Π j∈[p↓1]t j Π i∈[1↑g][x i, y i]> which permute the set of conjugacy classes {[t j], [t̄ j]: j ∈ [1↑[1↑p]}. It now follows from a result of Nielsen that M algg, p is generated by the image of the ADLH set together with a reflection. This gives a new way of seeing that M algg, p equals the (topological) mappingclass group of S g,p, along lines suggested by Magnus, Karrass, and Solitar in 1966. © de Gruyter 2011.