On the geometry of the symmetrized bidisc

We study the action of the automorphism group of the $2$ complex dimensional manifold symmetrized bidisc $\mathbb{G}$ on itself. The automorphism group is 3 real dimensional. It foliates $\mathbb{G}$ into leaves all of which are 3 real dimensional hypersurfaces except one, viz., the royal variety. This leads us to investigate Isaev’s classification of all Kobayashi-hyperbolic 2 complex dimensional manifolds for which the group of holomorphic automorphisms has real dimension 3 studied by Isaev. Indeed, we produce a biholomorphism between the symmetrized bidisc and the domain $$\{(z_1,z_2)\in {\mathbb{C}}^2:1+\mathrm{\mid }{z}_1{\mathrm{\mid }}^2-\mathrm{\mid }{z}_2{\mathrm{\mid }}^2>\mathrm{\mid }1+z_1^2-z_2^2\mathrm{\mid },Im(z_1(1+\overline{z_2}))>0\}$$ in Isaev’s list. Isaev calls it $\mathcal D_1$. The road to the biholomorphism is paved with various geometric insights about $\mathbb{G}$. Several consequences of the biholomorphism follow including two new characterizations of the symmetrized bidisc and several new characterizations of ${\mathcal{D}}_1$. Among the results on ${\mathcal{D}}_1$, of particular interest is the fact that ${\mathcal{D}}_1$ is a “symmetrization”. When we symmetrize (appropriately defined in the context in the last section) either ${\mathrm{\Omega }}_1$ or ${\mathcal{D}}_1^{(2)}$ (Isaev’s notation), we get ${\mathcal{D}}_1$. These two domains ${\mathrm{\Omega }}_1$ and ${\mathcal{D}}_1^{(2)}$ are in Isaev’s list and he mentioned that these are biholomorphic to $\mathbb{D}\times \mathbb{D}$. We produce explicit biholomorphisms between these domains and $$\mathbb{D}$ \times $\mathbb{D}$$.