THE LARGE-SCALE GEOMETRY OF HILBERT MODULAR-GROUPS

Let G be the rank-2 semisimple Lie group PSL(2)(R) x PSL(2)(R). In thi s paper we give a canonical isomorphism between the quasi-isometry gro up and the commensurator group of an irreducible, nonuniform lattice i n G. The most familiar of these lattices are the classical Hilbert mod ular groups PSL(2)(O-d), where O-d is the ring of integers in the real quadratic field Q(root d). As corollaries to this theorem we obtain t he following results: 1. The complete quasi-isometry classification of lattices in G. 2. Let Gamma be any finitely generated group. If Gamma is quasi-isometric to an irreducible, nonuniform lattice Lambda in G, then Gamma is a finite extension of an irreducible, nonuniform lattic e commensurable with Lambda in G. 3. Two irreducible, nonuniform latti ces in G are quasi-isometric iff they are commensurable. In particular , no two distinct classical Hilbert modular groups are quasi-isometric .