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**4.3 PROOF OF FINITE GENERATION**

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To show that Mod(*S*) is finitely generated we consider its action on complex (*S*). Note that Mod(*S*) indeed acts on (*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 (*S*).

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