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Abstract: Hilbert-space shattering is usually presented as a delicate phenomenon, wherein generic perturbations quickly restore ergodicity. In this work we introduce a large class of models exhibiting robust ergodicity breaking in quantum dynamics. The shattering in these models is topological in the sense that it persists to all orders of perturbation theory. This means that ergodicity is not restored, at least for timescales exponentially long in the perturbation strength. The models have crisp connections to gauge theories, and generalize Kitaev's quantum double to infinite groups. We also argue that in some group-based models, within a region of parameter space, each Krylov sector hosts a lowest-energy state that remains absolutely stable to generic perturbations for times that diverge with system size. We discuss how this conjectured absolute stability depends on the input group. In three dimensions we argue that this ergodicity breaking even survives coupling to a nonzero-temperature heat bath.