Free abelian quotients of commensurators

Abstract

For every group $\Gamma$, we define a homomorphism $d^\Gamma : \mathrm{Comm}(\Gamma) \to \mathbb{Z}^{(\mathcal{FS})}$ from the abstract commensurator $\mathrm{Comm}(\Gamma)$ to the free abelian group $\mathbb{Z}^{(\mathcal{FS})}$ with basis the collection $\mathcal{FS}$ of isomorphism classes of finite simple groups. We investigate the homomorphism $d^\Gamma$ when $\Gamma$ is a finitely generated free group $F$. We explicitly describe the image of $d^F : \mathrm{Comm}(F) \to \mathbb{Z}^{(\mathcal{FS})}$, which is a free abelian group of infinite rank, and we show that the kernel is the monolith of the group $\mathrm{Comm}(F)$. We deduce in particular that every proper quotient of $\mathrm{Comm}(F)$ is abelian. We use this to study the commensurator of a cocompact lattice in the automorphism group of a regular tree, which can be seen as a subgroup of $\mathrm{Comm}(F)$. We show that the image of this group under $d^F$ is again a free abelian group of infinite rank, showing in particular this group is not virtually simple.