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Let $\Gamma$ be either the infinite cyclic group $\mathbb{Z}$ or theBaumslag-Solitar group $\mathbb{Z} \ltimes \mathbb{Z}[\frac{1}{2}]$. Let $K$ bea slice knot admitting a slice disc $D$ in the 4-ball whose exterior hasfundamental group $\Gamma$. We classify the $\Gamma$-homotopy ribbon slicediscs for $K$ up to topological ambient isotopy rel. boundary. In the infinitecyclic case, there is a unique equivalence class of such slice discs. When$\Gamma$ is the Baumslag-Solitar group, there are at most two equivalenceclasses of $\Gamma$-homotopy ribbon discs, and at most one such slice disc foreach lagrangian of the Blanchfield pairing of $K$.

Characterisation of homotopy ribbon discs