The large-scale geometry of Hilbert modular groups

Let G be the rank-2 semisimple Lie group PSL2(R) x PSL 2 {R).In this paper we give a canonical isomorphism between the quasi-isometry group and the commensurator group of an irreducible, nonuniform lattice in G.The most familiar of these lattices are the classical Hubert modular groups PSL,2(Pd), where Od is the ring of integers in the real quadratic field Q(y/d).As corollaries to this theorem we obtain the following results:1.The complete quasi-isometry classification of lattices in G.2. Let Γ be any finitely generated group.If Γ is quasi-isometric to an irreducible, nonuniform lattice Λ in G, then Γ is a finite extension of an irreducible, nonuniform lattice commensurable with Λ in G.3. Two irreducible, nonuniform lattices in G are quasi-isometric iff they are commensurable.In particular, no two distinct classical Hubert modular groups are quasi-isometric.