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

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