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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.

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

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