Twisted Local Wild Mapping Class Groups: Configuration Spaces, Fission Trees and Complex Braids

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 the following: (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 result known as “Riemann’s count” of the dimensions of the moduli spaces of compact Riemann surfaces.