If $\mathbf{G}$ is an algebraic group defined over $\mathbb{Q}$, a subgroup of $\mathbf{G}(\mathbb{Q})$ is *arithmetic* if it is commensurable to $\mathbf{G}(\mathbb{Q}) \cap \operatorname{GL}\_n(\mathbb{Z})$ where some representation $\mathbf{G} < \operatorname{GL}\_n$ has been chosen (and the definition is made so that the choice does not matter).


Fur the purpose of this question let us call a subgroup $\Gamma$ of $\mathbf{G}(\mathbb{Q})$ *strictly arithmetic* if there exists a group $\mathbb{Z}$-scheme $\mathbf{G}\_\mathbb{Z}$ with generic fiber $\mathbf{G}$ such that $\Gamma = \mathbf{G}\_\mathbb{Z}(\mathbb{Z})$.


I was recently asked the natural question whether strictly arithmetic is the same as arithmetic. I suspect that the answer is "no". More specifically arithmetic groups can be arbitrarily small (for instance have arbitrarily large covolume in $\mathbf{G}(\mathbb{R})$) while I suspect that this is not true of strictly arithmetic groups. But I don't know enough about group schemes to underpin that intuition. So I'm asking here:


**Original question: Are there (resp. what are) examples of arithmetic groups that are not strictly arithmetic?**


The original question was answered in the comments by David Loeffler using a different obstruction so let me (following YCor's suggestion in the comments) specifically ask:


**Additional question: Do there exist "arbitrarily small" strictly arithmetic subgroups, for instance in the sense that the covolume or injectivity radius in $\mathbf{G}(\mathbb{R})$ is arbitrarily large?**



Arithmetic groups and integral points of integral structures
**First question** (do non-strictly-arithmetic subgroups exist?):


Any "strictly arithmetic" subgroup in your sense will, in particular, be a congruence subgroup, i.e. the intersection of $G(\mathbb{Q})$ with an open compact subgroup in $G(\mathbb{A}\_f)$. Since non-congruence subgroups exist in $SL\_2 / \mathbb{Q}$, and in lots of other groups too, these are examples of arithmetic subgroups which are not strictly arithmetic.


**Second question** (can strictly arithmetic subgroups be small?):


Start with your favourite $GL\_n$-embedding $\iota$, defining some strictly arithmetic $\Gamma$. Take some $g \in G(\mathbb{Q})$ which isn't in $\Gamma$, and consider the embedding into $GL\_{2n}$ sending $h$ to the block-diagonal matrix $\begin{pmatrix} \iota(h) \\& \iota(g^{-1} h g))\end{pmatrix}$. This defines a new $\mathbb{Z}$-model of $G$ whose integral points are $\Gamma \cap g \Gamma g^{-1}$. If $G = SL\_2$, and probably for just about any $G$ which isn't abelian, the index of $\Gamma \cap g \Gamma g^{-1}$ in $\Gamma$ can be made arbitrarily large by a suitable choice of $\Gamma$.


