Exponential factorizations of holomorphic maps

We show that any element of the special linear group $S{L}_2(R)$ is a product of two exponentials if the ring $R$ is either the ring of holomorphic functions on an open Riemann surface or the disc algebra. This is sharp: one exponential factor is not enough since the exponential map corresponding to $S{L}_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $G{L}_2(R)$.