Characteristic polynomials of pseudo-Anosov maps

We study the relationship between three different approaches to the action of a pseudo-Anosov mapping class [F ] on a surface: the original theorem of Thurston, its algorithmic proof by Bestvina-Handel, and related investigations of Penner-Harer. Bestvina and Handel represent [F ] as a suitably chosen homotopy equivalence f : G → G of a finite graph, with an associated transition matrix T whose largest eigenvalue is the dilatation of [F ]. Extending a skew-symmetric form introduced by Penner and Harer to the setting of Bestvina and Handel, we show that the characteristic polynomial of T is a monic and palindromic or anti-palindromic polynomial, possibly multiplied by a power of x. Moreover, it factors as a product of three polynomials. One of them reflects the action of [F ] on a certain symplectic space, the second one relates to the degeneracies of the skew-symmetric form, and the third one reflects the restriction of f to the vertices of G. We give an application to the problem of deciding whether certain transition matrices are induced by a pseudo-Anosov mapping class.