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4. Generating the Mapping Class Group > 4.3 Proof of Finite Generation - Pg. 104

104 CHAPTER 4 1 Z 1 Z Mod(S ) 1 1 (U T ( S)) Mod( S) 1 1 1 ( S) Mod( S, p) Mod( S) 1 1 1 To get the middle row directly, one can consider the fiber bundle Diff + ( S, (p, v)) Diff + ( S) U T ( S). 4.3 PROOF OF FINITE GENERATION To show that Mod(S) is finitely generated we consider its action on complex N (S). Note that Mod(S) indeed acts on N (S) since homeomorphisms take nonseparating simple closed curves to nonseparating simple closed curves and homeomorphisms preserve geometric intersection number. It is a basic principle from geometric group theory that if a group G acts cellularly on a connected cell complex X and if D is a subcomplex of X whose G- translates cover X, then G is generated by the set {g G : gD D = } (this idea will be echoed in our proof of Theorem 8.2 below). The next lemma is a specialized version of this fact designed specifically so that we can apply it to the action of Mod(S) on N (S). Lemma 4.10 Suppose that a group G acts by simplicial automorphisms on a connected, 1-dimensional simplicial complex X. Suppose that G acts transitively on the vertices of X and that it also acts transitively on pairs of vertices of X that are connected by an edge. Let v and w be two vertices of X that are connected by an edge and choose h G so that h(w) = v. Then the group G is generated by the element h together with the stabilizer of v in G. Proof. Let g G. We would like to show that g is contained in the subgroup