Stratified braid groups I: monodromy

Abstract

The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. This is the first of a planned series of articles on the topology of these strata. At the center of our analysis is a study of the infinite-area translation surface associated to the logarithmic derivative $df/f$ of the polynomial. Here we determine the monodromy of these strata in the braid group, thus describing which braidings of the roots are possible if the orders of the critical points are required to stay fixed. Mirroring the story for holomorphic differentials on higher-genus surfaces, we find the answer is governed by the framing of the punctured disk induced by the horizontal foliation on the translation surface.Comment: 28 pages, 9 figures. Comments welcome