Polynomial invariants of pseudo-Anosov maps (Q2885379)

In [Topology 34, No. 1, 109--140 (1995; Zbl 0837.57010)], \textit{M. Bestvina} and \textit{M. Handel} gave an algorithmic proof of Thurston's classification theorem for mapping classes (see e.g., the volume [Travaux de Thurston sur les surfaces. Seminaire Orsay. Astérisque, No. 66--67. Paris: Société Mathématique de France. 284 p. (1979; Zbl 0406.00016)]. If \([F]\) is a pseudo-Anosov map acting on an orientable surface \(S\), their algorithm allows to construct a graph \(G\) (homotopic to \(S\) when \(S\) is punctured), a suitable map \(f: G \to G\) (called \textit{train track map}) and the associated \textit{transition matrix} \(T\) (whose \textit{Perron-Frobenius eigenvalue} is the \textit{dilatation} of \([F]\); see [\textit{F. R. Gantmacher}, The theory of matrices, Vol. 2. Providence, RI: AMS Chelsea Publishing. (1959; Zbl 0927.15002)]).NEWLINENEWLINEThe dilatation \(\lambda(F)\) is an invariant of the conjugacy class \([F]\) in the modular group of \(S\), studied by \textit{C. T. McMullen} [Ann. Sci. Éc. Norm. Supér. (4) 33, No. 4, 519--560 (2000; Zbl 1013.57010)] and in several subsequent papers.NEWLINENEWLINEThe present paper introduces a new approach to the study of invariants of \([F]\), when \([F]\) is pseudo-Anosov: starting from the Bestvina-Handel algorithm, the authors investigate the structure of the characteristic polynomial of the transition matrix \(T\) and obtain two new integer polynomials (both containing \(\lambda(F)\) as their largest real root), which turn out to be invariants of the given pseudo-Anosov mapping class.NEWLINENEWLINEThe degrees of these new polynomials, as well as of their product, are invariants of [F], too; simple formulas are given for computing them by a counting argument from an invariant train track.NEWLINENEWLINEThe paper gives also examples of genus 2 pseudo-Anosov maps having the same dilatation, which are distinguished by the new invariants.