The group Aut and Out of the fundamental group of a closed Sol 3-manifold

Abstract

Let $E$ be the fundamental group of a closed Sol 3-manifold. We describe the groups $Aut(E)$ and $Out(E)$. We first consider the case where $E$ is the fundamental group of a torus bundle, and then the case where $E$ is the fundamental group of a closed Sol 3-manifold which is not a torus bundle. The groups are described in terms of some iterated semi-direct products of well known groups, where one exception is for $Out(E)$ for some $E$'s, in which case $Out(E)$ is described as an extension and a presentation is given.