# Exponential factorizations of holomorphic maps

## TL;DR

This paper proves that elements of SL_2(R) over certain rings can be expressed as products of two exponentials, extending to GL_2(R), with implications for the surjectivity of the exponential map.

## Contribution

It establishes the minimal number of exponentials needed to factor elements of SL_2(R) over rings of holomorphic functions or the disc algebra, extending to GL_2(R).

## Key findings

- Any element of SL_2(R) is a product of two exponentials.
- One exponential factor is insufficient due to non-surjectivity.
- The result applies to rings of holomorphic functions and the disc algebra.

## Abstract

We show that any element of the special linear group $SL_2(R)$ 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 $SL_2(\mathbb{C})$ is not surjective. Our result extends to the linear group $GL_2(R)$.