REVIEW OF: "Birman Joan - Brinkmann Peter - Kawamuro Keiko, Polynomial invariants of pseudo-Anosov maps, J. Topol. Anal. 4, No. 1, 13-47 (2012)". [DE060376660]

In [Topology 34, No.1, 109-140 (1995; Zbl 0837.57010)], Bestvina and Handel gave an algorithmic proof of Thurston’s classification theorem for mapping classes (see e.g., [Astrisque, 66-67 (1979; Zbl 0446.57005-23)]). If [F] is a pseudo-Anisov 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 ! G (called train track map) and the associated transition matrix T (whose Perron-Frobenius eigenvalue is the dilatation of [F]: see [The theory of matrices, Vol. 2, AMS Chelsea Publishing (1959; Zbl 0927.15002)]). The dilatation (F) is an invariant of the conjugacy class [F] in the modular group of S, studied in [Ann. Sci. c. Norm. Supr. (4) 33, No. 4, 519-560 (2000; Zbl 1013.57010)] and in several subsequent papers. The present paper introduces a new approach to the study of invariants of [F], when [F] is pseudo-Anisov: starting from 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 (F) as their largest real root), which turn out to be invariants of the given pseudo-Anisov mapping class. The 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. The paper gives also examples of genus 2 pseudo-Anisov maps having the same dilatation, which are distinguished by the new invariants.