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.