Characteristic polynomials of pseudo-Anosov maps

Peter Brinkmann (CCNY)

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 $f: G\to G$ 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 clas s. This is joint work with Joan Birman and Keiko Kawamuro.