Affine group as topological group

Question

I have a rather simple question. How one can think about discrete subgroups of affine group? That is my question is about explicit topology on the affine group.

Any links and comments will be helpful.

Many thanks.

Answer 1

The group ${\rm Aff} (\mathbb{R}^n)$ is a generalization of the isometry group of $\mathbb{R}^n$. It can be represented by matrices, namely by $$ {\rm Aff} (\mathbb{R}^n)=\left\{\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix} \mid A\in GL_n(\mathbb{R}), b\in \mathbb{R}^n\right\}. $$ Since matrix groups are Lie groups, we can apply the concept of discrete subgroups of Lie groups.

Answer 2

Affine group is composed of matrices, thus has a natural topology (namely the subspace topology from $\mathbb R^{n^2}$).