Abstract commensurators of lattices in Lie groups

Abstract

Let $\Gamma$ be a lattice in a simply-connected solvable Lie group. We construct a $\mathbb{Q}$-defined algebraic group $\mathcal{A}$ such that the abstract commensurator of $\Gamma$ is isomorphic to $\mathcal{A}(\mathbb{Q})$ and Aut$(\Gamma )$ is commensurable with $\mathcal{A}(\mathbb{Z})$. Our proof uses the algebraic hull construction, due to Mostow, to define an algebraic group $\mathbf{H}$ so that commensurations of $\Gamma$ extend to $\mathbb{Q}$-defined automorphisms of $\mathbf{H}$. We prove an analogous result for lattices in connected linear Lie groups whose semisimple quotient satisfies superrigidity.