# Criteria for Zariski density of subgroups of reductive groups

Let $$G$$ be a reductive group over a number field $$K$$. Let $$\Gamma\subset G(K)$$ be a subgroup.

My extremely naive question is - When can you deduce that $$\Gamma$$ is Zariski-dense? I'm looking for criteria for Zariski-density, possibly making additional assumptions on $$G,K,\Gamma$$.

To my surprise I've found it to be very difficult to find information on this question. References would also be appreciated!

As a special case of this, let $$R_K\subset K$$ be the ring of integers. Fix an embedding of algebraic $$K$$-groups $$i : G\hookrightarrow \text{GL}_{n,K}$$, and define $$G(R_K) := \text{GL}_n(R_K)\cap i(G(K))$$. When is $$G(R_K)$$ Zariski-dense? Does it depend on the embedding $$i$$?

It might be appropriate to make this a community wiki

The groups $$G(R_K)$$ you are interested in are arithmetic groups. If $$G$$ is semisimple, then a theorem of Borel and Harish-Chandra says that $$G(R_K)$$ is a lattice in $$G(K \otimes \mathbb{R})$$. Modulo some other minor hypotheses (connected, no compact factors) you can then appeal to the Borel Density Theorem to see that $$G(R_K)$$ is Zariski dense in $$G(K \otimes \mathbb{R})$$. A nice source that discusses this kind of thing is Dave Witte-Morris's book "Introduction to Arithmetic groups", available here
For this, you definitely need $$G$$ to be semisimple. For instance, $$GL(n,\mathbb{Z})$$ is not Zariski dense in $$GL(n,\mathbb{R})$$ (and is also not a lattice).
• Do you mean to say that $G(R_K)$ is Zariski dense in $G$? The Borel density theorem seems to only address the case that $K$ can be embedded in $\mathbb{R}$. What if $K$ is imaginary? Jan 5, 2022 at 5:46
• @stupid_question_bot: Use Weil Restriction is to reduce to the case $K=\mathbb{Q}$. en.m.wikipedia.org/wiki/Weil_restriction Jan 5, 2022 at 11:53