  We show that in closed string topology and in open-closed string topology
with one $D$-brane, higher genus stable string operations are trivial. This is
a consequence of Harer's stability theorem and related stability results on the
homology of mapping class groups of surfaces with boundaries. In fact, this
vanishing result is a special case of a general result which applies to all
homological conformal field theories with a property that in the associated
topological quantum field theories, the string operations associated to genus
one cobordisms with one or two boundaries vanish. In closed string topology,
the base manifold can be either finite dimensional, or infinite dimensional
with finite dimensional cohomology for its based loop space. The above
vanishing result is based on the triviality of string operations associated to
homology classes of mapping class groups which are in the image of stabilizing
maps.
