Cyclotomic polynomials and homological criteria for mapping class types

Abstract

Let $S_g$ be a closed orientable surface and let $\Psi\colon \mathrm{Mod}(S_g)\to \mathrm{Sp}(2g,\mathbb{Z})$ 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 $n\geq 3$, the polynomial $\varphi_n(x)$ 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 $\varphi_n(x)$ 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.