Alex Bene
University of Southern California

Lifting the Magnus Representation to the Mapping Class Groupoid

Abstract: Reporting on joint work with Shigeyuki Morita and Robert Penner, I will discuss how the Magnus representation of the mapping class group relative to an embedded disk lifts to the corresponding mapping class groupoid. This lifting makes essential use of the cell decomposition of Teichm\"uller space described in terms of decorated fatgraphs; generators of the groupoid correspond to Whitehead moves on such fatgraphs, and to each such move we associate a Fox Jacobian matrix. These matrices are determined solely by the combinatorics and decoration of the graphs, and fall into 36 basic classes independent of genus. Simple modifications of this construction yield similar representations of the higher Torelli groupoids and lead to practical algorithms for computing the higher Johnson homomorphisms.