Monodromy of stratified braid groups, II

Abstract The space of monic squarefree polynomials has a stratification according to the multiplicities of the critical points, called the equicritical stratification. Tracking the positions of roots and critical points, there is a map from the fundamental group of a stratum into a braid group. We give a complete determination of this map. It turns out to be characterized by the geometry of the translation surface structure on $$\mathbb{C}\mathbb{P}^1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msup> </mml:mrow> </mml:math> induced by the logarithmic derivative $$\textrm{d}f/f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>d</mml:mtext> <mml:mi>f</mml:mi> <mml:mo>/</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> of a polynomial in the stratum.