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

Abstract

Let [Formula: see text] be the fundamental group of a closed [Formula: see text] [Formula: see text]-manifold. We describe the groups [Formula: see text] and [Formula: see text]. We first consider the case where [Formula: see text] is the fundamental group of a torus bundle, and then the case where [Formula: see text] is the fundamental group of a closed [Formula: see text] [Formula: see text]-manifold which is not a torus bundle. The groups are described in terms of some iterated semidirect products of well-known groups, where one exception is for [Formula: see text] for some [Formula: see text], in which case [Formula: see text] is described as an extension and a presentation is given.