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).
Categories: math.CV