On the Andreadakis–Johnson filtration of the automorphism group of a free group

Families of non-trivial cohomology classes are given for the discrete groups belonging to the Johnson filtration of the automorphism group of a free group generated by n letters. The methods are (1) to analyze analogous classes for filtrations of a subgroup of the pure symmetric automorphism group of a free group and (2) to analyze features of these classes which are preserved by the Johnson homomorphism with values in the Lie algebra of derivations of a free Lie algebra. One consequence is that the ranks of the cohomology groups in any fixed degree i for 1⩽i⩽n−2 for the Johnson filtrations of IAn increase without bound. The actual classes constructed are fragile in the sense that they vanish after passage to successive filtrations; furthermore, these classes are all naturally in the image of the Johnson homomorphism on the level of cohomology. The methods are similar to those occurring within the theory of arrangements.The authors congratulate Corrado De Concini on the occasion of his 60-th birthday. One of the authors would like to thank Corrado for a wonderful time arising from a discussion about cohomology of groups with friends in the basement of Il Palazzone!