# Classification of elements in mapping class groups

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?

• 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. – Ryan Budney Nov 3 '14 at 2:56
• Any references for the tools you mentioned? – YCC Nov 3 '14 at 3:00
• see also flipper by Mark Bell: front.math.ucdavis.edu/1410.1358 bitbucket.org/Mark_Bell/flipper – Ian Agol Nov 3 '14 at 3:46

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