  We study a C*-dynamical system arising from the ring inclusion of the 2\times
2 integer matrices in the rational ones. The orientation preserving affine
groups of these rings form a Hecke pair that is closely related to a recent
construction of Connes and Marcolli; our dynamical system consists of the
associated reduced Hecke C*-algebra endowed with a canonical dynamics defined
in terms of the determinant function. We show that the Schlichting completion
also consists of affine groups of matrices, over the finite adeles, and we
obtain results about the structure and induced representations of the Hecke
C*-algebra. In a somewhat unexpected parallel with the one dimensional case
studied by Bost and Connes, there is a group of symmetries given by an action
of the finite integral ideles, and the corresponding fixed point algebra
decomposes as a tensor product over the primes. This decomposition allows us to
obtain a complete description of a natural class of equilibrium states which
conjecturally includes all KMS_\beta-states for \beta\ne 0,1.
