Commensurators and deficiency

J.A.Hillman

Abstract

We show that if G is a finitely generated group the kernel of the natural homomorphism from G to its abstract commensurator Comm(G) is locally nilpotent by locally finite, and is finite if G has deficiency >1. We also give a simple proof that the commensurator of SL(n,\mathbb{Z}) in GL(n,\mathbb{R}) is generated by GL(n,\mathbb{Q}) and scalar matrices.

Keywords: commensurable. deficiency. volume condition.

AMS Subject Classification: Primary 20F28;; secondary 20F99.