Free abelian quotients of commensurators

For every group $Γ$, we define a homomorphism $d^Γ: \mathrm{Comm}(Γ) \to \mathbb{Z}^{(\mathcal{FS})}$ from the abstract commensurator $\mathrm{Comm}(Γ)$ to the free abelian group $\mathbb{Z}^{(\mathcal{FS})}$ with basis the collection $\mathcal{FS}$ of isomorphism classes of finite simple groups. We