Abstract commensurators of lattices in Lie groups

TL;DR

This paper constructs an algebraic group to describe the abstract commensurators of lattices in solvable and certain linear Lie groups, linking group automorphisms to algebraic group rational points.

Contribution

It introduces a method to realize the abstract commensurator of such lattices as rational points of a Q-defined algebraic group, extending previous algebraic hull techniques.

Findings

Abstract commensurators are isomorphic to A(Q) for a constructed algebraic group A.

Automorphism groups are commensurable with A(Z).

Results apply to lattices in solvable and certain linear Lie groups with superrigidity.

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.