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

Mardi, 27. novembre 2018 - 14:00 - 15:00

Daciberg Gonçalves (São Paulo)


The closed $3$-manifolds are divided into 8 families characterized by a geometry on $\mathbb{R}^3$. Then we focus in one of the families called Sol manifolds, and we explain the topological properties and the construction of this family. 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 the middle term of certain extensions:

$1\to H \to G\to \mathbb{Z}_2\to 1$, where $H\in F_0$ (*).  

The goal is to present results about $Aut(G)$, as well as $Out(G)$, 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 corresponds 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 characteristic kernel. 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, such as properties of the group $GL(2,Z)$. One of the conclusions is that we will obtain a short exact sequence of the form $1\to \mathbb{Z}+\mathbb{Z} \to Aut(G) \to \mathbb{Z}_2+(\mathbb{Z}\rtimes \mathbb{Z}_2)\to 1$ for $G\in F_1$.


This is joint work with Sergio Martins, Federal University of Santa Catarina-Florianópolis-Brazil