  We extend the notion of multi-moment map to geometries defined by closed
forms of arbitrary degree. We give fundamental existence and uniqueness results
and discuss a number of essential examples, including geometries related to
special holonomy. For forms of degree four, multi-moment maps are guaranteed to
exist and are unique when the symmetry group is (3,4)-trivial, meaning that the
group is connected and the third and fourth Lie algebra Betti numbers vanish.
We give a structural description of some classes of (3,4)-trivial algebras and
provide a number of examples.
