# Classification discrete subgroups of the general affine group.

Let $V$ be a real vector space of dimension $n$ and consider $GL(V)\ltimes V$ the general affine group of $V$.

I would like to know about the classification of discrete subgroups of $GL(V)\ltimes V$.

I am also interested in to know if there is an example of a discrete subgroup $H$ of $GL(V)\ltimes V$ such that the $H$-orbits of the natural action of $H$ on $V$ are non-discrete sets.

Thank you in advance for any reference.

• I doubt you can get much of a classification, since for example all finite groups of order $n$ are discrete subgroups. Sep 4 '18 at 17:58

It is unclear what do you mean by a "classification". For instance, the group $$SL(2, {\mathbb R})$$ already contains continuum of pairwise non-isomorphic discrete subgroups (countable free products of finite cyclic groups). If you want a discrete subgroup with nondiscrete orbits, consider the following example: $${\mathbb Z}^2 \rtimes SL(2, {\mathbb Z}) < {\mathbb R}^2 \rtimes SL(2, {\mathbb R}).$$ This discrete subgroup acts on $$V={\mathbb R}^2$$ so that almost orbit is dense in $$V$$. (This follows by considering the standard action of $$SL(2, {\mathbb Z})$$ on the torus $$T^2$$: Almost all orbits are dense.) There are probably examples of discrete groups of affine transformations where every orbit is dense, but I do not see a sufficient motivation for constructing such subgroups.