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Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids

Philip Boalch, Jean Douçot, Gabriele Rembado

TL;DR

<3-5 sentence high-level summary>

Abstract

Following the completion of the algebraic construction of the Poisson wild character varieties (B.--Yamakawa, 2015) one can consider their natural deformations, generalising both the mapping class group actions on the usual (tame) character varieties, and the G-braid groups already known to occur in the wild/irregular setting. Here we study these wild mapping class groups in the case of arbitrary formal structure in type A. As we will recall, this story is most naturally phrased in terms of admissible deformations of wild Riemann surfaces. The main results are: 1) the construction of configuration spaces containing all possible local deformations, 2) the definition of a combinatorial object, the ``fission forest'', of any wild Riemann surface and a proof that it gives a sharp parameterisation of all the admissible deformation classes. As an application of 1), by considering basic examples, we show that the braid groups of all the complex reflection groups known as the generalised symmetric groups appear as wild mapping class groups. As an application of 2), we compute the dimensions of all the (global) moduli spaces of type A wild Riemann surfaces (in fixed admissible deformation classes), a generalisation of the famous ``Riemann's count'' of the dimensions of the moduli spaces of compact Riemann surfaces.

Twisted local wild mapping class groups: configuration spaces, fission trees and complex braids

TL;DR

<3-5 sentence high-level summary>

Abstract

Following the completion of the algebraic construction of the Poisson wild character varieties (B.--Yamakawa, 2015) one can consider their natural deformations, generalising both the mapping class group actions on the usual (tame) character varieties, and the G-braid groups already known to occur in the wild/irregular setting. Here we study these wild mapping class groups in the case of arbitrary formal structure in type A. As we will recall, this story is most naturally phrased in terms of admissible deformations of wild Riemann surfaces. The main results are: 1) the construction of configuration spaces containing all possible local deformations, 2) the definition of a combinatorial object, the ``fission forest'', of any wild Riemann surface and a proof that it gives a sharp parameterisation of all the admissible deformation classes. As an application of 1), by considering basic examples, we show that the braid groups of all the complex reflection groups known as the generalised symmetric groups appear as wild mapping class groups. As an application of 2), we compute the dimensions of all the (global) moduli spaces of type A wild Riemann surfaces (in fixed admissible deformation classes), a generalisation of the famous ``Riemann's count'' of the dimensions of the moduli spaces of compact Riemann surfaces.
Paper Structure (30 sections, 32 theorems, 104 equations, 1 figure)

This paper contains 30 sections, 32 theorems, 104 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Layout of the paper
  3. General setting
  4. Twisted irregular types
  5. The exponential local system
  6. Irregular classes, finite subcovers and levels
  7. Irregular types
  8. Pullback to untwisted case
  9. Admissible deformations
  10. Numerical equivalence of irregular types
  11. Configuration spaces
  12. Classification of admissible deformations
  13. A single Stokes circle
  14. Single circle configuration spaces
  15. The case of two Stokes circles
  16. ...and 15 more sections

Key Result

Theorem 1.1

Two irregular classes $\Theta,\Theta'$ at a point $a\in \Sigma$ are admissible deformations of each other if and only if their fission trees are isomorphic: $\mathcal{T}(\Theta)\cong \mathcal{T}(\Theta')$. Consequently two rank $n$ wild Riemann surfaces ${\bf \Sigma}, {\bf \Sigma}'$ are admissible d

Figures (1)

  • Figure 1: The fission tree $\mathcal{T}^\flat$ associated to $Q$ (not drawn isometrically). The labelling corresponds to the numbering of the $q_i$. The multiplicities of the leaves are all equal to $1$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 64 more