Deciding reducibility of mapping classes is in $\textbf{NP}$

Abstract

For a fixed marked surface $S$, we show that the problem of deciding whether or not a mapping class is reducible lies in $\textbf{NP}$. As usual this immediately gives an exponential time algorithm to decide whether or not a mapping class is reducible. To do this we use an (ideal) triangulation to obtain a coordinate system on the set of multicurves on $S$. The result then follows from the fact that the action of the mapping class group of $S$ is piecewise-linear with respect to such a coordinate system and so we are able so show that: if a mapping class $h$ fixes a multicurve then it fixes one whose size is at most exponential in the word length of $h$. We go on to show how to repeat this construction on invariant subsurfaces. This allows us to show that a similar bound holds for the size of the canonical curve system of a mapping class and so give an alternate, elementary proof of a result of Koberda and Mangahas.