## 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.

: Primary 20F28;; secondary 20F99.

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 Monday, July 30, 2007