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

TL;DR

This paper explores wild mapping class groups associated with wild Riemann surfaces, introducing configuration spaces and fission forests to parameterize deformations, and linking these groups to complex braid groups and moduli space dimensions.

Contribution

It constructs configuration spaces for wild Riemann surfaces, defines fission forests for classifying deformations, and connects wild mapping class groups to complex braid groups and moduli space calculations.

Findings

Wild mapping class groups include complex reflection group braid groups.

Configuration spaces encompass all local deformations of wild Riemann surfaces.

Dimensions of moduli spaces are computed, generalizing Riemann's classical results.

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.