Much ado about no offset -- Characterising the anomalous multiple-image configuration and the model-driven displacement between light and mass in the multi-plane strong lens Abell 3827

Abell 3827 is a unique galaxy cluster with a dry merger in its core causing a highly-resolved multiple-image configuration of a blue spiral galaxy at $z_\mathrm{s}=1.24$. The surface brightness profiles of four merging galaxies around $z_\mathrm{d}=0.099$ complicate a clear identification of the number of images and finding corresponding small-scale features across them. The entailed controversies about offsets between luminous and dark matter have never been settled and dark-matter characteristics in tension with bounds from complementary probes and simulations seemed necessary to explain this multiple-image configuration. We resolve these issues with a systematic study of possible feature matchings across all images and their impact on the reconstructed mass density distribution. From the local lens properties directly constrained by these feature matchings without imposing any global lens model, we conclude that none of them are consistent with expected local characteristics from standard single-lens-plane lensing, nor can they be motivated by the light distribution in the cluster. Inspecting complementary spectroscopic data, we show that all these results originate from an insufficient constraining power of the data and seem to hint at a thick lens and not at exotic forms of dark matter or modified gravity. If the thick-lens hypothesis can be corroborated with follow-up multi-plane lens modelling, A3827 suffers from a full three-dimensional degeneracy in the distribution of dark matter because combinations of shearings and scalings in a single lens plane can also be represented by an effective shearing and a rotation caused by multiple lens planes.