Filtrations on the Mapping Class Group of the punctured sphere
Moduli and Friends seminar
Monday 28 February 2022
Jacques Darné (UCLouvain) — Filtrations on the Mapping Class Group of the punctured sphere
The Mapping Class Group of the n-punctured sphere (fixing a basepoint) classically identifies with the quotient Bn* of the Artin braid group by its center. This group acts faithfully on the fundamental group of the the n-punctured sphere, which is free on n-1 generators. From this action, one can define a filtration of Bn* by subgroups Bn* ⊇ A1 ⊇ A2 ⊇ ⋯, which are the analogues of the Johnson kernels in this situation. It turns out that this filtration is exactly the lower central series of the corresponding pure mapping class group Pn*, and that we understand fairly well the associated Lie algebra, using Milnor invariants adapted to these mapping classes. In this talk, I intend to explain these results, and to give you a taste of the techniques involved in their proofs.
* Eastern European Time, i.e. UTC+2 in winter and UTC+3 in summer.