Commensurators of solvable $S$-arithmetic groups

Daniel Studenmund

Vol. 283 (2016), No. 1, 201–221

DOI: 10.2140/pjm.2016.283.201

Abstract

We show that the abstract commensurator of an $S$-arithmetic subgroup of a solvable algebraic group over $\mathbb{Q}$ is isomorphic to the $\mathbb{Q}$-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 $\left(\mathbb{Z}∕2\mathbb{Z}\right)\wr \mathbb{Z}$.

Keywords

solvable groups, arithmetic groups, commensurator, abstract commensurator, lamplighter group

Mathematical Subject Classification 2010

Primary: 20G30

Secondary: 11H56, 20E36

Milestones

Received: 3 April 2014

Revised: 1 October 2015

Accepted: 7 October 2015

Published: 14 June 2016