  We make a connection between the structure of the bidisc and a distinguished
subgroup of its automorphism group. The automorphism group of the bidisc, as we
know, is of dimension six and acts transitively. We observe that it contains a
subgroup that is isomorphic to the automorphism group of the open unit disc and
this subgroup partitions the bidisc into a complex curve and a family of
strongly pseudo-convex hypersurfaces that are non-spherical as CR-manifolds.
Our work reverses this process and shows that any $2$-dimensional
Kobayashi-hyperbolic manifold whose automorphism group (which is known, from
the general theory, to be a Lie group) has a $3$-dimensional subgroup that is
non-solvable (as a Lie group) and that acts on the manifold to produce a
collection of orbits possessing essentially the characteristics of the
concretely known collection of orbits mentioned above, is biholomorphic to the
bidisc. The distinguished subgroup is interesting in its own right. It turns
out that if we consider any subdomain of the bidisc that is a union of a proper
sub-collection of the collection of orbits mentioned above, then the
automorphism group of this subdomain can be expressed very simply in terms of
this distinguished subgroup.
