This is a survey of results on arithmetic groups. Only a minimal acquaintance with algebraic geometry is assumed. Theorems are mostly quoted without proof: sometimes an indication of the method is given. There is a substantial bibliography, with a guide to the subjects it covers. The reader is referred to the sources therein for the proofs omitted here.

DEFINITIONS AND GENERAL PROPERTIES ([9], [26])

Let G be an algebraic subgroup of GLn, defined over the field Q of rational numbers. Thus there exists a set of polynomials (with rational coefficients) in the n2 matrix entries and the inverse of the determinant, whose set of solutions in any extension E of Q is a subgroup GE, of GLn (E). We call GE the group of E-points of G. The groups GR and Gc are respectively real and complex Lie groups.

We write GZ for GQ ∩ GLn (Z).

Definition. A subgroup Г of GQ is arithmetic if it is commensurable with GZ: that is, if Г ∩GZ has finite index both in Г and in Gz.

A group Г is arithmetic if it can be embedded as an arithmetic subgroup in GQ for some Q-algebraic subgroup G of GLn. Then any subgroup of finite index in Г is also an arithmetic group.

Remarks. (1) We admit all subgroups commensurable with Gz, rather than Gz alone, in order to make the definition independent of the chosen Q-embedding of G in a general linear group.