Let \(M\) be a connected complex manifold of dimension \(n\) and denote by \(\operatorname{Aut}(M)\) the group of holomorphic automorphisms of \(M\). If \(M\) is Kobayashi hyperbolic then \(\operatorname{Aut}(M)\) carries the structure of a Lie group. It is known that \(d(M)\), the dimension of \(\operatorname{Aut}(M)\) is upper bounded by \(n^2+2n\) with equality if and only if \(M\) is holomorphically equivalent to the unit ball.
\(M\) is homogeneous whenever \(d(M)> n^2\). If \(d(M)=n^2\) then \(M\) need not to be homogeneous, futher not all hyperbolic manifolds with \(d(M)=n^2\) are complete and simply connected.
In this paper the author classifies connected hyperbolic manifolds of dimension \(n\geq 2\) greater or equal two with \(d(M))=n^2\) (up to holomorphic equivalence). In the proof, one can find the respective automorphism group.