  In this article we deduce some algebraic properties for the group
$\mathrm{Sp}_{2n} (\mathcal{O}(X))$ of holomorphic symplectic matrices on a
Stein space $X$: holomorphic factorization, exponential factorization, and
Kazhdan's property (T). In holomorphic factorization we combine a recent result
of the third author and K-theory tools to give explicit bounds for the case
when $X$ is one-dimensional or two-dimensional. Next we use them to find bounds
for exponential factorization. As a further application, we show that the
elementary symplectic group $\mathrm{Ep}_{2n}(\mathcal{O}(X))$ admits Kazhdan's
property (T).

Complex Variables