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A Primer on Mapping Class Groups (PMS-49) by Dan Margalit, Benson Farb

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Chapter Four

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Generating the Mapping Class Group

Is there a way to generate all (homotopy classes of) homeomorphisms of a surface by compositions of simple-to-understand homeomorphisms? We have already seen that Mod(T2) is generated by the Dehn twists about the latitude and longitude curves. Our next main goal will be to prove the following result.

THEOREM 4.1 (Dehn–Lickorish theorem) For g ≥ 0, the mapping class group Mod(Sg) is generated by finitely many Dehn twists about nonseparating simple closed curves.

Theorem 4.1 can be likened to the theorem that for each n ≥ 2 the group SL(n, ) can be generated by finitely many elementary matrices. As ...

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