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

4.3 PROOF OF FINITE GENERATION

To show that Mod(S) is finitely generated we consider its action on complex ncap(S). Note that Mod(S) indeed acts on ncap(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 euro G : gD common17 Dzeroslash} (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 ncap(S).

Lemma 4.10 Suppose that a group G acts by simplicial automorphisms on a connected, 1-


  

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