We show that the Nielsen–Thurston classification of mapping classes of the sphere with four marked points is determined by the quantum $SU(n)$ representations, for any fixed $n\geq 2$. In the Pseudo–Anosov case we also show that the stretching factor is a limit of eigenvalues of (non-unitary) $SU(2)$-TQFT representation matrices. It follows that at big enough levels, Pseudo–Anosov mapping classes are represented by matrices of infinite order.
Math. Proc. Camb. Phil. Soc, 2006, Vol 141, p. 477-488