  We study gradient Ricci solitons with maximal symmetry. First we show that
there are no non-trivial homogeneous gradient Ricci solitons. Thus the most
symmetry one can expect is an isometric cohomogeneity one group action. Many
examples of cohomogeneity one gradient solitons have been constructed. However,
we apply the main result in our paper "Rigidity of gradient Ricci solitons" to
show that there are no noncompact cohomogeneity one shrinking gradient solitons
with nonnegative curvature.
