Paper detail

An effective algebraic detection of the Nielsen--Thurston classification of mapping classes

In this article, we propose two algorithms for determining the Nielsen-Thurston classification of a mapping class $ψ$ on a surface $S$. We start with a finite generating set $X$ for the mapping class group and a word $ψ$ in $\langle X \rangle$. We show that if $ψ$ represents a reducible mapping class in $\Mod(S)$ then $ψ$ admits a canonical reduction system whose total length is exponential in the word length of $ψ$. We use this fact to find the canonical reduction system of $ψ$. We also prove an effective conjugacy separability result for $π_1(S)$ which allows us to lift the action of $ψ$ to a finite cover $\yt{S}$ of $S$ whose degree depends computably on the word length of $ψ$, and to use the homology action of $ψ$ on $H_1(\yt{S},\mathbb{C})$ to determine the Nielsen-Thurston classification of $ψ$.