Assume that $G$ is a semi-simple linear algebraic group defined over $\mathbb{Q}$, which is $\mathbb{Q}$-simple, and that $G(\mathbb(R)$ is non-compact, without $\mathbb{R}$-factors of rank 1. Then by Margulis's works, the arithmetic subgroups of $G(\mathbb{R})$ are the same as discrete lattice in $G(\mathbb{R})$. Here a lattice is a discrete subgroup $\Gamma$ such that the quotient $\Gamma\backslash G(\mathbb{R})$ is of finite volume with respect to the measure deduced from the left Haar measure. In particular, an arithmetic subgroup in $G(\mathbb{R})$ is Zariski dense in $G_\mathbb{R}$.
Conversely, a discrete subgroup
$\Gamma$ in $G(\mathbb{R})$
is given, such that $\Gamma$ is also dense in $G_\mathbb{R}$ for the Zariski topology, what condition should one impose to make it arithmetic? Shall I assume $\Gamma$ to be finitely generated, or stable under certain actions such as $Aut(\mathbb{R/Q})$? I feel that such kind of results are more or less available in the literature, bu I'm far from an expert in this field.
Many thanks!
Best Answer
Arbitrary Zariski-dense subgroups in a semisimple group can be very small from a real-analytic point of view. It seems that algebra cannot distinguish between "small" and "large" Zariski-dense subgroups, so most criteria to distinguish between the two have a strong non-algebraic flavour. (Of course one can also characterize arithmetic groups algebraically, but this has even less to do with the line of argument you seem to suggest.) From a dynamical point of view, the key difference between lattices and arbitrary Zariski-dense subgroups is that the former act transitively on the product of the Furstenberg boundary of the ambient Lie group with itself ("double ergodicity"). This is a sort of "largeness" property. There are various ways to capture this property, the most systematic way seems to me the concept of a generalized Weyl group due to Bader and Furman.