On page 101 of this paper of Laurent Bartholdi (which is an online documentation of the FR package for GAP which allows GAP to manipulate groups generated by automata) he gives a different description of the Baumslag-Solitar group
gap> A := FRAffineGroup(1,Integers,3);
<self-similar group over [ 1 .. 3 ]>
gap> f := Correspondence(A);
MappingByFunction( ( Integers^[ 2, 2 ] ), <self-similar group over [ 1 .. 3 ]>,
function( mat ) ... end )
gap> BaumslagSolitar := Group([[2,0],[0,1]]^f,[[1,0],[1,1]]^f);
<self-similar group over [ 1 .. 3 ] with 2 generators>"
This description of the Baumslag-Solitar group is quite unusual. How can I relate this definition of Bartholdi with the classical one? I didn't also find anything relevant about the relation between affine groups and Baumslag-Solitar groups. Bartholdi states thatt the group $\mathbb{Z}[1/2]\rtimes_2\mathbb{Z}$ is the Baumslag Solitar described by the above GAP procedure. Why? What means the $\rtimes_2$? I have always seen a Baumslag-Solitar group as this $$BS(1,m)=\langle a,b| bab^{-1}=a^m\rangle.$$
These Baumslag-Solitar groups BS(1,m) are linear: view a=[1,1;0,1] and b=[m,0;0,1].
Finitely generated linear groups can be represented using finite automata, rather than matrices. Explicit automata are described in
http://arxiv.org/abs/math/0603032
and they are those that are implemented in GAP.