Commensurators of solvable S-arithmetic groups

TL;DR

This paper characterizes the abstract commensurators of S-arithmetic subgroups in solvable algebraic groups over Q, revealing their algebraic structure and contrasting with nonlinear examples in positive characteristic, including the lamplighter group.

Contribution

It establishes that the abstract commensurator of such groups is isomorphic to the Q-points of an algebraic group, providing new insights into their algebraic structure.

Findings

Abstract commensurator is isomorphic to Q-points of an algebraic group

Comparison with nonlinear commensurators in positive characteristic

Description of the commensurator of the lamplighter group

Abstract

We show that the abstract commensurator of an S-arithmetic subgroup of a solvable algebraic group over Q is isomorphic to the Q-points of an algebraic group, and compare this with examples of nonlinear abstract commensurators of S-arithmetic groups in positive characteristic. In particular, we include a description of the abstract commensurator of the lamplighter group.