arithmetic groups VS. Zariski dense discrete subgroups? - MathOverflow

arithmetic groups VS. Zariski dense discrete subgroups?

2010-11-17T14:40:12Z

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!

Answer by Keivan Karai for arithmetic groups VS. Zariski dense discrete subgroups?

2010-11-17T15:11:29Z

an arithmetic subgroup is a lattice. This is in characteristic zero due to Borel and Raghunathan (MR0147566) and in positive characteristic due to Harder and Behr.

Answer by Tobias Hartnick for arithmetic groups VS. Zariski dense discrete subgroups?

2010-11-18T10:39:02Z

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.