Alladi Ramakrishnan Hall

Automorphisms of the fundamental group of a $3$-dimensional Sol-manifold

Daciberg Lima Goncalves

Institute of Mathematics and Statistics, University of Sao Paulo, Brazil

We begin by give a quick and simplified overview of the closed $3$-manifolds, divided into 8 families. Then we focus in one of the families called Sol-manifolds. In order to describe this family we consider the following two types of groups:

$F_0$ the family of groups of the form $(\mathbb{Z}+\mathbb{Z})\rtimes_{\theta}\mathbb{Z}$
where $\theta(1)\in Gl(2,\mathbb{Z})$ is an Anosov matrix, and
$F_1$ the groups which are middle term of certain extension
$1\to H \to G\to \mathbb{Z}_2\to 1$ where $H\in F_0$ (*).
The goal is to present results about $Aut(G)$, as well $Out(G)$,
for $G\in F_i$ for $i=0,1$.
We will explain the relation of these groups with the
3-manifolds, called Sol-Manifolds. As one motivation, the solution of this problem corresponde to the description of the homeomorphism of the manifold up to isotopy. The general approach is to study
automorphisms of extensions (short exact sequence) with kernel characteristic. For the groups in $F_0$
we use the sequence $1\to \mathbb{Z}+\mathbb{Z} \to G\to \mathbb{Z}\to 1$ and for the groups in $F_1$ we use the short exact sequence (*) above. The description of the groups will be obtained using several facts,
geometric and algebraic, notoriously properties of the group
$Gl(2,Z)$. One of the conclusion is that we will obtain a short exact sequence:
$$1\to \mathbb{Z}+\mathbb{Z} \to Aut(G) \to \mathbb{Z}_2+(\mathbb{Z}\rtimes \mathbb{Z}_2)\to 1$$
for $G\in F_1$.