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

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) rankG 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).