A hypersurface in $\mathbb {C}^2$ whose stability group is not determined by $2$-jets

Abstract:

We give an example of a hypersurface in $\mathbb {C}^2$ through $0$ whose stability group at $0$ is determined by $3$-jets, but not by jets of any lesser order. We also examine some of the properties which the stability group of this infinite type hypersurface shares with the $3$-sphere in $\mathbb {C}^2$.