Arithmeticity of rank-1 lattices with dense commensurators in positive characteristic

✍ Scribed by Lucy Lifschitz

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 126 KB
• Volume: 261
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00670-1

✦ Synopsis

G. Margulis showed that if G is a semisimple Lie group and Γ ⊂ G is an irreducible lattice, which has an infinite index in its commensurator, and which satisfies one of the following conditions:

(1) it is cocompact; (2) at least one of the simple components of G is defined over a local field of characteristic 0; (3) rank G 2, then Γ is arithmetic. This leaves out the case of non-uniform lattices in rank-1 simple groups G defined over a local field of positive characteristic. We show the arithmeticity of the lattice Γ in this remaining case (under the assumption of density of its commensurator).