Canonical extensions of the Johnson homomorphisms to the Torelli groupoid

We prove that every trivalent marked bordered fatgraph comes equipped with a canonical generalized Magnus expansion in the sense of Kawazumi. This Magnus expansion is used to give canonical extensions of the higher Johnson homomorphisms τm, for m⩾1, to the Torelli groupoid, and we provide a recursive combinatorial formula for tensor representatives of these extensions. In particular, we give an explicit 1-cocycle in the dual fatgraph complex which extends τ2 and thus answer affirmatively a question of Morita and Penner. To illustrate our techniques for calculating higher Johnson homomorphisms in general, we give explicit examples calculating τm, for m⩽3.