  In this paper, we mainly study geometric constructions of thin Blaschke
products $B$ and reducing subspace problem of multiplication operators induced
by such symbols $B$ on the Bergman space. Considering such multiplication
operators $M_B$, we present a representation of those operators commuting with
both $M_B$ and $M_B^*$. It is shown that for "most" thin Blaschke products $B$,
$M_B$ is irreducible, i.e. $M_B$ has no nontrivial reducing subspace; and such
a thin Blaschke product $B$ is constructed.
  As an application of the methods, it is proved that for "most" finite
Blaschke products $\phi$, $M_\phi$ has exactly two minimal reducing subspaces.
Furthermore, under a mild condition, we get a geometric characterization for
when $M_B$ defined by a thin Blaschke product $B$ has a nontrivial reducing
subspace.
