Abstract commensurators of lattices in Lie groups

Abstract

Let Gamma be a lattice in a simply-connected solvable Lie group. We construct a Q-defined algebraic group A such that the abstract commensurator of Gamma is isomorphic to A(Q) and Aut(Gamma) is commensurable with A(Z). Our proof uses the algebraic hull construction, due to Mostow, to define an algebraic group H so that commensurations of Gamma extend to Q-defined automorphisms of H. We prove an analogous result for lattices in connected linear Lie groups whose semisimple quotient satisfies superrigidity.Comment: 35 pages. v4 added references and improved exposition based on referee's comments. Added results 6.4 and 6.5 relating abstract commensurator of lattice to automorphism group of some ambient Lie group. Final version, to appear in Comm. Math. Hel