Cyclotomic polynomials and homological criteria for mapping class types

Thomas Koberda; Åukasz Patryk Michalak

geometric topology

Cyclotomic polynomials and homological criteria for mapping class types

arXiv:2607.14599

summary

The paper characterizes which cyclotomic polynomials can appear as characteristic polynomials of integral symplectic matrices coming from algebraically finite type mapping classes, showing this occurs only when the order has at most two distinct prime factors, and uses this to give a homological criterion for pseudo‑Anosov mapping classes.

Abstract

Let $S_{g}$ be a closed orientable surface and let $I^¨ : Mod (S_{g}) \to Sp (2 g, Z)$ 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 $I ¨ _{n} (x)$ 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 $I ¨ _{n} (x)$ 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.

17 pages, one figure

Topics & keywords

#mapping class groups #cyclotomic polynomials #symplectic matrices #pseudo-Anosov #homological criteriaΦ_n(x)algebraically finite typepseudo-AnosovSp(2g,ℤ)characteristic polynomialcyclotomic