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

TL;DR

This paper characterizes the automorphism and outer automorphism groups of the fundamental group of closed Sol 3-manifolds, providing explicit descriptions in various cases, including presentations and extensions.

Contribution

It offers a detailed description of Aut(E) and Out(E) for fundamental groups of closed Sol 3-manifolds, including cases beyond torus bundles, with explicit algebraic structures.

Findings

Aut(E) and Out(E) are described as iterated semi-direct products.

For some E, Out(E) is given as an extension with a presentation.

Explicit algebraic structures are provided for different classes of Sol 3-manifolds.

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.