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Local wild mapping class groups and cabled braids

Jean Douçot, Gabriele Rembado, Matteo Tamiozzo

TL;DR

The paper develops a comprehensive framework for local wild mapping class groups Γ_Q arising from isomonodromic deformations of irregular connections, proving a product decomposition and introducing fission trees to control their structure. In type A it identifies Γ_Q with pure cabled braid groups, providing an explicit operadic/cabling picture and detailed low-rank classifications; analogous, though more intricate, descriptions are given for types B, C, and D via bichromatic and generalized fission trees. The work also connects these groups to hyperplane arrangements and Stokes-data braiding, and illustrates the constructions with concrete type-A examples while outlining extensions to twisted and global settings. Overall, it lays the foundation for a multi-scale braid-theoretic description of local wild topology and its actions on wild character varieties.

Abstract

We will define and study some generalisations of pure $\mathfrak{g}$-braid groups that occur in the theory of connections on curves, for any complex reductive Lie algebra $\mathfrak{g}$. They make up local pieces of the wild mapping class groups, which are fundamental groups of (universal) deformations of wild Riemann surfaces, underlying the braiding of Stokes data and generalising the usual mapping class groups. We will establish a general product decomposition for the local wild mapping class groups, and in many cases define a fission tree controlling this decomposition. Further in type A we will show one obtains cabled versions of braid groups, related to braid operads.

Local wild mapping class groups and cabled braids

TL;DR

The paper develops a comprehensive framework for local wild mapping class groups Γ_Q arising from isomonodromic deformations of irregular connections, proving a product decomposition and introducing fission trees to control their structure. In type A it identifies Γ_Q with pure cabled braid groups, providing an explicit operadic/cabling picture and detailed low-rank classifications; analogous, though more intricate, descriptions are given for types B, C, and D via bichromatic and generalized fission trees. The work also connects these groups to hyperplane arrangements and Stokes-data braiding, and illustrates the constructions with concrete type-A examples while outlining extensions to twisted and global settings. Overall, it lays the foundation for a multi-scale braid-theoretic description of local wild topology and its actions on wild character varieties.

Abstract

We will define and study some generalisations of pure -braid groups that occur in the theory of connections on curves, for any complex reductive Lie algebra . They make up local pieces of the wild mapping class groups, which are fundamental groups of (universal) deformations of wild Riemann surfaces, underlying the braiding of Stokes data and generalising the usual mapping class groups. We will establish a general product decomposition for the local wild mapping class groups, and in many cases define a fission tree controlling this decomposition. Further in type A we will show one obtains cabled versions of braid groups, related to braid operads.
Paper Structure (38 sections, 23 theorems, 136 equations, 3 figures)

This paper contains 38 sections, 23 theorems, 136 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General aim
  3. Nonlinear monodromy actions in 2d gauge theory
  4. Main results and layout of the paper
  5. An example in type A
  6. Admissible deformations of wild Riemann surfaces
  7. Wild deformations
  8. Pure local wild mapping class groups
  9. Filtrations and fission
  10. Fission: Lie groups/algebras
  11. Fission: Dynkin diagrams
  12. General results
  13. Hyperplane arrangements
  14. Reduction to the simple case
  15. Nonempty complements
  16. ...and 23 more sections

Key Result

Theorem 1

The number of nontrivial factors of eq:pure_local_WMCG_factors-intro is at most the semisimple rank of $\mathfrak g$ (i.e., the rank of $\Phi_{\mathfrak g}$).

Figures (3)

  • Figure 1: Before cabling
  • Figure 2: Example of (pure) braid cabling
  • Figure 3: Example of fission tree (in type $A$)

Theorems & Definitions (73)

  • Theorem 1: § \ref{['sec:descending_ranks']}
  • Theorem 2: § \ref{['sec:low_rank']}
  • Theorem 3: Thmm. \ref{['thm:tree_gives_wmcg_type_A']}, \ref{['thm:tree_gives_wmcg_type_BC']} and \ref{['thm:tree_gives_wmcg_type_D']}
  • Theorem 4: Thm. \ref{['thm:wmcg_type_A_equal_cabled_braid_groups']}
  • Remark 2.1
  • Definition 2.1
  • Definition 2.2: boalch_2014_geometry_and_braiding_of_stokes_data_fission_and_wild_character_varieties, Def. 10.1
  • Remark 2.2
  • Definition 2.3: Cf. boalch_2014_geometry_and_braiding_of_stokes_data_fission_and_wild_character_varieties, Ex. 10.1
  • Proposition 2.1
  • ...and 63 more