The Nielsen-Thurston classification of mapping classes is determined by TQFT

Abstract

For each fixed $n \geq 2$ we show how the Nielsen-Thurston classification of mapping classes for a closed surface of genus $g \geq 2$ is determined by the sequence of quantum $SU(n)$-representations $(\rho_k)_{k \in {\mathbb{N}}}$. That this is the case is a consequence of the asymptotic faithfulness property proved in [A3]. We here provide explicit conditions on $(\rho_k (\phi))_{k\in {\mathbb{N}}}$, which determines the Nielsen-Thurston type of any mapping class $\phi$.