  Let l be a prime and G a pro-l group with torsion-free abelianization. We
produce group-theoretic analogues of the Johnson/Morita cocycle for G -- in the
case of surface groups, these cocycles appear to refine existing constructions
when l=2. We apply this to the pro-l etale fundamental groups of smooth curves
to obtain Galois-cohomological analogues, and discuss their relationship to
work of Hain and Matsumoto in the case the curve is proper. We analyze many of
the fundamental properties of these classes and use them to give an example of
a non-hyperelliptic curve whose Ceresa class has torsion image under the l-adic
Abel-Jacobi map.
