Arithmetic group

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In mathematics, an arithmetic group is a subgroup of a linear algebraic group with simple algebraic properties.

Formal definition[edit]

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 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 arithmetic groups and lattices in Lie groups 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.