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 Isaevs 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+|z_1|^2-|z_2|^2>|1+ z_1 ^2 -z_2 ^2|, Im(z_1 (1+overline{z_2}))>0}] in Isaevs 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 $Omega_1$ or $mathcal{D}^{(2)}_1$ (Isaevs notation), we get $mathcal D_1$. These two domains $Omega_1$ and $mathcal{D}^{(2)}_1$ are in Isaevs list and he mentioned that these are biholomorphic to $mathbb{D} times mathbb{D}$. We produce explicit biholomorphisms between these domains and $mathbb{D} times mathbb{D}$.