Dilatation of outer automorphisms of Right-angled Artin Groups

We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(\Gamma) = \langle V | E \rangle$andanautomorphism$\phi \in Out(A(\Gamma))$ there is a natural measure of how fast the length of a word of $A(\mathrm{\Gamma })$ grows after $n$ iterations of $\varphi$ as a function of $n$, which we call the dilatation of $w$ under $\varphi$. We define the dilatation of $\varphi$ as the supremum over dilatations of all $w\in A(\mathrm{\Gamma })$. Assuming that $\varphi$ is a pure and square map, we show that if the dilatation of $\varphi$ is positive, then either there exists a free abelian special subgroup on which that dilatation is realized; or there exists a strata of either free or free abelian groups on which the dilatation is realized.