SL(3,Z) is not Howson

We give an explicit construction of two $2$-generated subgroups $H,K\le \text{SL}(3,\mathbb{Z})$ whose intersection is not finitely generated. The construction takes place inside the standard parabolic subgroup ${\mathbb{Z}}^2⋊\text{SL}(2,\mathbb{Z})\le \text{SL}(3,\mathbb{Z})$. The main point is to identify $H\cap K$ with the stabilizer of a point for an affine action of a free group on ${\mathbb{Z}}^2$, and then to prove, using the Schreier graph of this action, that this stabilizer is not finitely generated. Furthermore, we prove that there exists a sequence of subgroups $H_q,K_q\le \text{SL}(3,\mathbb{Z})$ such that $\text{rank}(H_q)=\text{rank}(K_q)=4$, and $$\text{rank}(H_q\cap K_q)\ge q+1,$$ while $H_q\cap K_q$ is finitely generated.