Let $(\Gamma,\cdot,\leq)$ be an ordered abelian group. The convex rank of $\Gamma$ is the supremum of the length over chains of convex subgroups of $\Gamma$. A convex subgroup $H$ of $\Gamma$ is a subgroup such that for $x\in\Gamma,x'\in H$ with $1\geq x\geq x'$, we have $x\in H$.