Automorphisms of the Pants Complex

In loving memory of my mother, Batya 1. Introduction In the theory of mapping class groups, " curve complexes " assume a role similar to the one that buildings play in the theory of linear groups. Ivanov, Korkmaz, and Luo showed that the automorphism group of the curve complex for a surface is generally isomorphic to the extended mapping class group of the surface. In this paper, we show that the same is true for the pants complex. Throughout, S is an orientable surface whose Euler characteristic χ (S) is negative , while g,b denotes a surface of genus g with b boundary components. Also, Mod(S) means the extended mapping class group of S (the group of homotopy classes of self-homeomorphisms of S). The pants complex of S, denoted C P (S), has vertices representing pants decom-positions of S, edges connecting vertices whose pants decompositions differ by an elementary move, and 2-cells representing certain relations between elementary moves (see Sec. 2). Its 1-skeleton C 1 P (S) is called the pants graph and was introduced by Hatcher and Thurston. We give a detailed definition of the pants complex in Section 2. Brock proved that C 1 P (S) models the Teichmüller space endowed with the Weil-Petersson metric, T W P (S), in that the spaces are quasi-isometric (see [1]). Our results further indicate that C 1 P (S) is the " right " combinatorial model for T W P (S), in that Aut C 1 P (S) (the group of simplicial automorphisms of C 1 P (S)) is shown to be Mod(S). This is in consonance with the result of Masur and Wolf that the isometry group of T W P (S) is Mod(S) (see [10]).