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Recently I start learning mapping class group. The Nielsen-Thurston classification says that each element in mapping class group $Mod(S_{g,n}),g,n\geq 0$ is periodic, reducible, or pseudo-Anosov. Take any element in $Mod(S_{g,n})$, how to determine the element is in which one of the cases?

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  • $\begingroup$ Do you want an algorithm or would you be content with something more conceptual? As far as I know there aren't any super-fast algorithms but there's plenty of heuristic tools that get you nice places. $\endgroup$ – Ryan Budney Nov 3 '14 at 2:56
  • $\begingroup$ Any references for the tools you mentioned? $\endgroup$ – YCC Nov 3 '14 at 3:00
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    $\begingroup$ see also flipper by Mark Bell: front.math.ucdavis.edu/1410.1358 bitbucket.org/Mark_Bell/flipper $\endgroup$ – Ian Agol Nov 3 '14 at 3:46
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An effective algorithm can be found in

 Mladen Bestvina and Michael Handel. Train-tracks for surface homeomorphisms. Topology, vol. 34 (1995), no. 1, pp. 109–140

This has been implemented by Peter Brinkman in Xtrain (which you can get via computop.org).

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  • $\begingroup$ @Yu-ChanChang - If you like this answer, you should accept it (by clicking on the "check" mark.) $\endgroup$ – Sam Nead Apr 11 '16 at 11:48
  • $\begingroup$ @SamNead Does this person still exist? $\endgroup$ – Igor Rivin Apr 11 '16 at 11:58

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