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 BudneyCommented Nov 3, 2014 at 2:56
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$\begingroup$ Any references for the tools you mentioned? $\endgroup$– YCCCommented Nov 3, 2014 at 3:00
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5$\begingroup$ see also flipper by Mark Bell: front.math.ucdavis.edu/1410.1358 bitbucket.org/Mark_Bell/flipper $\endgroup$– Ian AgolCommented Nov 3, 2014 at 3:46
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1 Answer
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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).
answered Nov 3, 2014 at 3:05
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$\begingroup$ @Yu-ChanChang - If you like this answer, you should accept it (by clicking on the "check" mark.) $\endgroup$– Sam NeadCommented Apr 11, 2016 at 11:48
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$\begingroup$ @SamNead Does this person still exist? $\endgroup$ Commented Apr 11, 2016 at 11:58