Arithmetic Lattices
An archetypical example of a nonuniform lattice is given by the group SL(2,Z), which is a lattice in the special linear group SL(2,R), and by the closely related modular group. This construction admits a far-reaching generalization to a class of lattices in all semisimple algebraic groups over a local field F called arithmetic lattices. For example, let F = R be the field of real numbers. Roughly speaking, the Lie group G(R) is formed by all matrices with entries in R satisfying certain algebraic conditions, and by restricting the entries to the integers Z, one obtains a lattice G(Z). Conversely, Grigory Margulis proved that under certain assumptions on G, any lattice in it essentially arises in this way. This remarkable statement is known as Arithmeticity of lattices or Margulis Arithmeticity Theorem.
Read more about this topic: Lattice (discrete Subgroup)
Famous quotes containing the word arithmetic:
“I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.”
—Gottlob Frege (1848–1925)