Arithmetic group

Definition

Suppose $K$ is a number field and ${\mathcal {O}}$ is the ring of integers in $K$. Suppose $G$ is a linear algebraic group over $K$. An arithmetic subgroup of $G$ is a subgroup of $G$ that is commensurable with the group $G({\mathcal {O}})$. Here, $G({\mathcal {O}})$ refers to the subgroup of $G$ comprising those matrices such that both the matrix and its inverse have all entries in ${\mathcal {O}}$.

An arithmetic group is a group realized as an arithmetic subgroup in a linear algebraic group.