The Automorphism Group of the Complex of Pants Decompositions

There has been recent interest in finding combinatorial models for groups, i.e. finding a simplicial complex whose automorphism group is a given group. For example, CharneyDavis showed that the Coxeter Diagram is a model for Out(W ) (for certain Coxeter groups W) [3], and Bridson-Vogtmann showed that a spine of Outer Space is a model for Out(Fn) [1]. In this paper, we prove that the complex of pants decompositions is a model for the mapping class group. Throughout, S will denote a closed oriented surface of negative Euler characteristic. The complex of pants decompositions of S, defined by Hatcher-Thurston and denoted CP(S), has vertices representing pants decompositions of S, edges connecting vertices whose pants decompositions differ by an elementary move (see below), and 2-cells representing certain relations between elementary moves. Its 1-skeleton is called the graph of pants decompositions, and is denoted CP(S). Brock proved that CP(S) models the Teichmüller Space of S endowed with the WeilPetersson metric, T WP (S), in that the spaces are quasi-isometric [2]. Our main result further indicates that CP(S) is the “right” combinatorial model for T WP (S), in that the automorphism group of CP(S) is shown to be the extended mapping class group of S, Mod(S) (the group of diffeomorphisms of S to itself, modulo isotopy). This is in consonance with the result of Masur-Wolf that the isometry group of T WP (S) is Mod(S) [12]. The extended mapping class group has a natural action by automorphisms on CP(S); the content of the theorem is that all of the automorphisms of CP(S) are induced by elements of Mod(S). The Main Theorem of this paper is as follows: