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 \(\mathrm{Comm}(G)\) is locally nilpotent by locally finite, and is finite if \(G\) has deficiency \( > 1\). We also give a simple proof that the commensurator of \(\mathrm{SL}(n,\mathbb{Z})\) in \(\mathrm{GL}(n,\mathbb{R})\) is generated by \(\mathrm{GL}(n,\mathbb{Q})\) and scalar matrices.

Keywords: commensurable. deficiency. volume condition.

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