Simply-connected four-dimensional gradient Ricci solitons that are invariant under a compact cohomogeneity one group action have been studied extensively. However, the special case where the group is \(\textrm{SU}(2)\) (the smallest possible example) has received comparatively little attention. The purpose of this article is to give a comprehensive study of simply-connected \(\textrm{SU}(2)\) -invariant expanding cohomogeneity one gradient Ricci solitons. The main result is the construction of new 3-parameter families of complete \(\textrm{SU}(2)\) -invariant asymptotically conical expanding gradient Ricci solitons and new 2-parameter families of complete \(\textrm{SU}(2)\) -invariant Einstein metrics.
