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 $\psi$ on a surface $S$. We start with a finite generating set $X$ for the mapping class group and a word $\psi$ in $⟨X⟩$. We show that if $\psi$ represents a reducible mapping class in $\text{Mod}(S)$ then $\psi$ admits a canonical reduction system whose total length is exponential in the word length of $\psi$. We use this fact to find the canonical reduction system of $\psi$. We also prove an effective conjugacy separability result for ${\pi }_1(S)$ which allows us to lift the action of $\psi$ to a finite cover $\text{yt}S$ of $S$ whose degree depends computably on the word length of $\psi$, and to use the homology action of $\psi$ on $H_1(\text{yt}S,\mathbb{C})$ to determine the Nielsen-Thurston classification of $\psi$.