Thompson's group $F$ is not weakly amenable

Let $F$ be Thompson’s group, and let $D$ be the set of dyadic rationals in $[0,1]$. We give an embedding of ${\mathbb{Z}}^{(D)}⋊F$ into $F$, where the semidirect product is defined by the natural action of $F$ on $D$. Using this embedding, we deduce from the nonamenability of $F$ that $F$ is not weakly amenable.