Cyclotomic polynomials and homological criteria for mapping class types

Let $S_g$ be a closed orientable surface and let $\mathrm{\Psi }\text{\hspace{0.17em}⁣}:\mathrm{M}\mathrm{o}\mathrm{d}(S_g)\to \mathrm{S}\mathrm{p}(2g,\mathbb{Z})$ be the representation induced by the action on first homology. We investigate the characteristic polynomials of integral symplectic matrices arising from mapping classes of algebraically finite type and give a complete characterization in the cyclotomic case: for $n\ge 3$, the polynomial ${\phi }_n(x)$ is realized by a mapping class of algebraically finite type if and only if $n$ has at most two distinct prime divisors. Consequently, if $n$ is square-free and has at least three distinct prime divisors, then every mapping class with characteristic polynomial ${\phi }_n(x)$ is pseudo-Anosov. This gives a cyclotomic complement to the Casson–Bleiler homological criterion and yields a complete criterion for a symplectic polynomial to be realized only by pseudo-Anosov mapping classes.