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Let $X$ be a Riemann surface and $\Gamma$ its (pure) Mapping Class Group, then $\Gamma$ is generated by Dehn twists along simple closed curves. Is \emph{any} element of the mapping class group also a Dehn twists? Said in another way, any element of $\Gamma$ is some complicated word in the generators associated to some collection of pairwise disjoint closed curves. Does it also exist a curve such that this element is the twist associated to it?

P.S. I know that in general one has to take into account half-twists as well, but I am not thinking of that, and this is why I wrote "pure" Mapping Class Group

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    $\begingroup$ No. Nielson-Thurston gives a description of the type of elements you can have. If you know the description of mapping class group of torus it is pretty easy to produce finite, dehn twists, and Anosov mapping classes. Anyways this question isn't really research level. $\endgroup$
    – user35370
    Commented Aug 31, 2020 at 11:24
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    $\begingroup$ Every mapping class is a product of finitely many Dehn twists. This is Lickorish‘s Theorem. $\endgroup$
    – ThiKu
    Commented Aug 31, 2020 at 11:52
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    $\begingroup$ Another no: the map on homology induced by a Dehn twist has trace 2g. But there are lots of homeomorphisms where this trace is 0; for example interchange the two circle factors of a torus. $\endgroup$
    – Danny Ruberman
    Commented Aug 31, 2020 at 19:07

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