Local wild mapping class groups and cabled braids

We define and study some generalisations of pure <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> -braid groups, for any complex reductive Lie algebra <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝔤</mml:mi> </mml:math> . They naturally occur in the theory of isomonodromic deformations for meromorphic connections with irregular singularities on principal bundles over Riemann surfaces, covering the general untwisted case, going beyond the case of generic irregular types. These generalised braid groups make up local pieces of the wild mapping class groups, which in turn extend the usual mapping class groups and govern the braiding of Stokes data. We establish a general product decomposition for these local wild mapping class groups, and in all classical cases define a fission tree governing the decomposition; in particular in type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>D</mml:mi> </mml:math> we will find a factor which is not isomorphic to any pure braid group coming from a root system. In type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> </mml:math> , the fission tree and the pure braid group operad yields a proof of the corresponding “multi-scale” braiding conjecture.