Stratified braid groups: monodromy
Abstract
The space of monic squarefree complex polynomials has a stratification according to the multiplicities of the critical points. We introduce a method to study these strata by way of the infinite-area translation surface associated to the logarithmic derivative of the polynomial. 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.
Cite
@article{arxiv.2304.04627,
title = {Stratified braid groups: monodromy},
author = {Nick Salter},
journal= {arXiv preprint arXiv:2304.04627},
year = {2023}
}
Comments
New in V2: the main change is an expanded Section 4 that treats monodromy of both roots and critical points and includes a discussion of the "braided Gauss-Lucas theorem". Introduction, title, and abstract have been tweaked as well. One new figure