SL(3,Z) is not Howson

Abstract

We give an explicit construction of two $2$-generated subgroups $H,K\leq \SL(3,\Z)$ whose intersection is not finitely generated. The construction takes place inside the standard parabolic subgroup $\Z^2\rtimes \SL(2,\Z)\leq \SL(3,\Z)$. 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 $H_q, K_q \leq \SL(3,\mathbb{Z})$ such that $\rank(H_q)=\rank(K_q)=4$, and \[ \rank(H_q\cap K_q)\geq q+1, \] while $H_q\cap K_q$ is finitely generated.