Cyclotomic polynomials and homological criteria for mapping class types

2026/07/16 by Thomas Koberda, Łukasz Patryk Michalak
#math.GT #math.DS

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Abstract

Let S_g be a closed orientable surface and let Ψ\colon Mod(S_g)→ Sp(2g,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≥ 3, the polynomial φ_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 φ_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.

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