  We first show that for a bounded pseudoconvex domain with a manifold quotient
of finite-volume in the sense of Kahler-Einstein measure, the identity
component of the automorphism group of this domain is semi-simple without
compact factors. This partially answers an open question in [Fra95]. Then we
apply this result in different settings to solve several open problems, for
examples,
  (1). We prove that the automorphism group of the Griffiths domain [Gri71] in
$\mathbb{C}^2$ is discrete. This gives a complete answer to an open question
raised four decades ago.
  (2). We show that for a contractible HHR/USq complex manifold $D$ with a
finite-volume manifold quotient $M$, if $D$ contains a one-parameter group of
holomorphic automorphisms and the fundamental group of $M$ is irreducible, then
$D$ is biholomorphic to a bounded symmetric domain. This theorem can be viewed
as a finite-volume version of Nadel-Frankel's solution for the Kahzdan
conjecture, which has been open for years.
  (3). We show that for a bounded convex domain $D\subset \mathbb{C}^n$ of
$C^2$-smooth boundary, if $D$ has a finite-volume manifold quotient with an
irreducible fundamental group, then $D$ is biholomorphic to the unit ball in
$\mathbb{C}^n$, which partially solves an old conjecture of Yau.
  For (2) and (3) above, if the complex dimension is equal to $2$, more refined
results will be provided.
