In mathematics, an arithmetic group (arithmetic subgroup) in a linear algebraic group G defined over a number field K is a subgroup Γ of G(K) that is commensurable with G(O), where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them. It can be shown that this condition depends only on G, not on a given matrix representation of G.
Examples of arithmetic groups include therefore the groups GLn(Z). The idea of arithmetic group is closely related to that of lattice in a Lie group. Lattices in that sense tend to be arithmetic, except in well-defined circumstances. The exact relationship of the two concepts was established by the work of Margulis on superrigidity . The general theory of arithmetic groups was developed by Armand Borel and Harish-Chandra; the description of their fundamental domains was in classical terms the reduction theory of algebraic forms.