Arithmetic group

A subgroup $H$ of a linear algebraic group $G$ defined over the field $\mathbb{Q}$ of rational numbers, that satisfies the following condition: There exists a faithful rational representation $\rho : G \rightarrow \mathrm{GL}_n$ defined over $\mathbb{Q}$ (cf. Representation theory) such that $\rho(H)$ is commensurable with $\rho(G) \cap \mathrm{GL}(n,\mathbb{Z})$, where $\mathbb{Z}$ is the ring of integers (two subgroups $A$ and $B$ of a group $C$ are called commensurable if $A \cap B$ is of finite index in $A$ and in $B$). This condition is then also satisfied for any other faithful representation defined over $\mathbb{Q}$. More generally, an arithmetic group is a subgroup of an algebraic group $G$, defined over a global field $k$, that is commensurable with the group $G_O$ of $O$-points of $G$, where $O$ is the ring of integers of $k$. An arithmetic group $H \cap G_{\mathbb{R}}$ is a discrete subgroup of $G_{\mathbb{R}}$.

If $\phi : G \rightarrow G_1$ is a $k$-epimorphism of algebraic groups, then the image $\phi(H)$ of any arithmetic group $H \subset G$ is an arithmetic group in $G_1$ [1]. The name arithmetic group is sometimes also given to an abstract group that is isomorphic to an arithmetic subgroup of some algebraic group. Thus, if $k$ is an algebraic number field, the group $G_O \cong G'_{\mathbb{Z}}$, where $G'$ is obtained from $G$ by restricting the field of definition from $k$ to $\mathbb{Q}$, is called an arithmetic group. In the theory of Lie groups the name arithmetic subgroups is also given to images of arithmetic subgroups of the group of real points of $G_{\mathbb{R}}$ under the factorization of $G_{\mathbb{R}}$ by compact normal subgroups.

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Useful additional references are [a1]–[a3]. [a2] is an elementary introduction to the theory of arithmetic groups.

Conjectures of A. Selberg and I.I. Pyatetskii-Shapiro roughly state that for most semi-simple Lie groups discrete subgroups of finite co-volume are necessarily arithmetic. G.A. Margulis settled this question completely and, in particular, proved the conjectures in question. See Discrete subgroup for more detail.

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