8 - Selberg's Work on the Arithmeticity of Lattices and its Ramifications

Abstract

This chapter explores Selberg's work on the arithmeticity of lattices and its ramifications. Prior to 1960, it had been observed that the general constructions of lattices that were applicable in all dimensions were arithmetic. The strategy that was inspired by Selberg's 1960 paper for cocompact lattices, was applicable to non-cocompact lattices as well and touched off more ramifications. The conjecture of a stronger rigidity phenomenon arose from the attempt to fit Selberg's rigidity theorem, as generalized by Weil, into a topological transformation group setting. The proof of strong rigidity was completed in several stages. The arithmeticity theorem that was proved by Margulis, applies to the general problem of irreducible lattices contained in a direct product of semisimple algebraic groups.